Polkinghorne’s empirical evidence for god: math and a comprehensible universe

I wanted to put up a brief post to get reader reaction to two claims that are common among science-friendly theologians. They are a centerpiece, for instance, of John Polkinghorne’s arguments for God’s existence. In the evenings these days (to the detriment of my sleep), I’m reading Polkinghorne and Alvin Plantinga to learn about the brand of “sophisticated” theology that tries to reconcile faith and science.

I’ve already discussed Polkinghorne a few days ago, and the quotes I give below are from his book Science and Religion in Quest of Truth. As I noted in my earlier post, Polkinghorne, unlike many theologians, does think that pure revelation cannot suffice as evidence for God; one needs empirical observations as well.

I’ll give my brief reactions to the arguments, but, as I said, I’d like to know how readers would respond to these. I’m looking for serious responses, but feel free, as always, to be lighthearted as well.

His two arguments are these:

  • The universe can be comprehended by the rational faculties of humans, and its workings appear to adhere to laws of physics.
  • Much of our understanding of the universe is expressed through mathematics, which is “unreasonably effective” in encapsulating what we discover about nature. 

Ergo Jesus. 

Here are some supporting quotes from Polkinghorne, showing how he connects the above observations—both undoubtedly true—with God.

“. . . why is science possible at all in the deep way that has proved to be the case?” p. 71

“A distinguished nuclear physicist, Eugene Wigner, once asked, ‘Why is mathematics so unreasonably effective?’ Those seeking an understanding as complete as possible must ask what it could be that that links together the reason within (mathematical thinking) and the reason without (the structure of the physical world) in this remarkable way? The universe has not only proved to be astonishingly rationally beautiful, affording scientists the reward of wonder for all the labours of their research. Why are we so lucky?

It would surely be intolerably intellectually lazy not to seek to pursue this question.  Yet science itself will not provide its answer, for it is simply content to exploit the opportunities that these wonderful gifts afford us, without being in a position to explain their origin.  Theology, however, can step into the breach. Science has disclosed to us a world which, in its rational transparency and beauty, is shot through with signs of mind, and religious belief suggests that it is indeed the Mind of the Creator that lies behind the wonderful order of the universe. (p. 73)

I have several tentative responses, and I believe Sean Carroll has addressed some of the above, though I can’t immediately lay my hands on his essays and posts (perhaps he’ll weigh in here).

  • If the Universe didn’t obey physical laws, we wouldn’t exist, for the universe we know couldn’t have formed in the first place.  Too, we couldn’t exist as biological entities if the physiology and biochemistry of our bodies didn’t adhere to physical laws—natural selection could not create faculties and senses that behave with regularity.
  • If the Universe didn’t always obey physical laws, that, too, could be taken as evidence for God, who could intercede with alarming regularity to alter whatever laws did exist.  Heads God wins, tails physicists lose.
  • We don’t understand why there are physical laws that behave with regularity, but that lack of understanding doesn’t point to the existence of a creator—much less Polkinghorne’s Christian creator who birthed Jebus.
  • The existence of those laws must (like the existence of God itself to theologians) be left as something that does not require an answer—or the answer that “that’s just the way things are.” I think this is physicist Sean Carroll’s answer.
  • Physical laws could differ—or even be irregular—in other universes that may exist. (Remember that multiverses are not, as some theologians imply, a “desperation move” on the part of physicists to explain fine-tuning and the anthropic principle. Rather, multiverses fall out as a prediction of some theories of physics.)
  • These are god-of-the-gap arguments: science could one day explain the existence of regular physical laws, but right now we don’t understand them, or why they must take the form they do.
  • In other places Polkinghorne (along with other theologians) argue that there are many phenomena in nature that cannot be comprehended with science, much less mathematics. These supposedly include love, aesthetics, and morality.  Those phenomena, too, are given as evidence for God. In other words, Polkinghorne is trying to have it both ways: the universe’s comprehensibility via science is given as evidence for God, but aspects that supposedly aren’t scientifically comprehensible are also given as evidence for god.
  • The “God” explanation offered by Polkinghorne is not testable, that is, it can’t be disconfirmed. Even if we find out why there are laws of physics, Polkinghorne could argue that God was behind it all.

I know that there are a fair number of physics- and math-savvy readers here, and I’m particularly interested in their responses (particularly about the “unreasonable effectiveness of mathematics”), but anybody is welcome to respond.

269 Comments

  1. Posted February 23, 2012 at 5:33 am | Permalink

    What you say about regularity is surely the key issue. As soon as any regularity appears, descriptive laws can be proposed. I would think a truly chaotic reality is a physical impossibility, and maybe a logical impossibility – wouldn’t the chaotic nature of reality be a law of its own? I’d have to think about that. Anyway, in either case, there would be no ‘why question’ to answer for regularities that can be described.

    Even if there were a ‘why question’ to answer, Polknghorne’s “Theology, however, can step into the breach” is unhelpful, since so can ‘making stuff up’ step into the breach. In fact, that is what happens, it seems.

    • truthspeaker
      Posted February 23, 2012 at 10:00 am | Permalink

      In some of Michael Moorcock’s fiction, where gods of Law and Chaos are adversaries, one character visits a plane ruled entirely by Chaos and discovers that it is entirely featureless.

      • Kharamatha
        Posted February 24, 2012 at 8:25 am | Permalink

        Which happens to be the ultimate in uniform order.

  2. Posted February 23, 2012 at 5:50 am | Permalink

    Mathematics is, at its origin, just a model.

    It started with counting. Then as people noticed patterns in math *that corresponded to patterns in nature*, we got algebra. Calculus sprang from a need to model physical phenomena that couldn’t otherwise be modelled.

    It may have become its own discipline, but let’s keep in mind that its historical foundations are as a modelling language for an objective universe.

    Math isn’t “unreasonably effective,” it’s just as effective as we made it. And being a model I would say it’s not *unreasonably* effective at all because it is intractable to represent mathematically *and very precisely* almost any real system.

    As with all theists, Polkinghorne is blinded to the facts by his fairy tale god.

    • Heinrich Kruger
      Posted February 23, 2012 at 7:43 am | Permalink

      Exactly this! We developed mathematics as a tool to describe and model the world. Therefore it should really come as no surprise that it is so effective at describing and modelling the real world.

      • dale
        Posted February 23, 2012 at 9:38 am | Permalink

        And until the advent of computers many of the models had to be linear in nature. Now math is starting to understand the three body problem. I would call that “unreasonably effective”, in the sense that at this point in time it really is unreasonable to expect math models to do a perfect job mimicking the world around us. One thing I can guarantee about any math model of the world around us is that they are wrong.

        • infiniteimprobabilit
          Posted February 24, 2012 at 12:11 am | Permalink

          If you try writing a program to model any real-world feature or system you soon realise the limitations of mathematics! Approximations everywhere.

    • clive photo
      Posted February 23, 2012 at 7:58 am | Permalink

      Seconded

    • Posted February 23, 2012 at 9:11 am | Permalink

      Exactly, if there were really an intelligence at work weaving the fabric of reality, then pi would be 3 and circles wouldn’t make my brain hurt.

      • Posted February 23, 2012 at 9:18 am | Permalink

        Not to mention the Pi button on my calculator would deliver up something altogether more delicious.

      • McWaffle
        Posted February 23, 2012 at 12:58 pm | Permalink

        I was just thinking about this exact same thing. If math is so perfect, what exactly does the 10^10^10^10th digit of Pi signify in reality? If you were calculating the area of a circle and approximated pi to the googleplexth digit, rather than the googleplexth +1 digit, what would be the difference of area? It would be something so arbitrarily miniscule as to be completely meaningless. And even then, you’ve still got an infinity of futher decimal places you’re ignoring!

        Perhaps somebody with the time and mathematics/physics chops could do an estimate. How many digits of Pi do you have to go down before the difference it makes comes to less than one Planck length?

        • Posted February 23, 2012 at 1:02 pm | Permalink

          You can work it out for yourself without much trouble. It’s been ages since I’ve done so, myself, but I seem to recall that the Earth’s orbit down to an inch or so only needs about a half dozen digits, and the observable universe down to Planck length only needs a few dozen digits.

          Cheers,

          b&

          • Kharamatha
            Posted February 24, 2012 at 8:28 am | Permalink

            Excellent, I can handle a half dozen.

        • Posted February 24, 2012 at 1:31 pm | Permalink

          That would be the gap where God built his condo.

        • Posted February 24, 2012 at 4:07 pm | Permalink

          In itself, nothing. But to me, a mathematical fictionalist, math in itself is the most sublime and difficult of arts, an intensely creative activity limited by almost nothing. (Even consistency is not an absolute criterion; there are paraconsistent mathematical theories.) Consequently the ability to calculate pi in that realm of fiction is a minor source of beauty.

          In the world itself, nothing at all, if you don’t think the world is continuous.

      • LOL
        Posted June 27, 2012 at 5:08 am | Permalink

        The comma was not invented in mathematics until 700 years ago, so anything before that had to be rounded up.

    • Posted February 23, 2012 at 11:32 am | Permalink

      + 1

    • Kevin Alexander
      Posted February 23, 2012 at 11:50 am | Permalink

      I agree only I would call it an analog computer. We invent machines (which is what maths are) that parallel some regular aspect of the observable universe.
      Newton invented a beautifully elegant machine that almost perfectly parallels the action of gravity. Then someone else invented a more perfect machine using different gears.

    • Kiwi Dave
      Posted February 23, 2012 at 9:30 pm | Permalink

      My thoughts exactly.

      The word ‘unreasonably’ seems to be doing a lot of heavy lifting in this claim. What exactly does it mean, and if it were deleted would the claim seem as persuasive?

      Language,if a bit fuzzy, also does quite an effective job of describing the universe for the lay person,including those things which mathematical science doesn’t describe very well, but no one claims it as evidence of divinity, I suppose.

    • Sj
      Posted March 24, 2012 at 7:08 am | Permalink

      The point is that the reality of math exists whether we learn it or not. We didn’t invent math, we merely figured it out. And there’s so much we still don’t know.

      • Posted March 26, 2012 at 1:54 pm | Permalink

        With respect, I disagree. Point me out an integer and I’ll change my mind.

  3. Manfredi La Manna
    Posted February 23, 2012 at 5:57 am | Permalink

    By giving attention to the Polkinghornes of this world we give credibility to what is essentially a fraudulent sleight-of-hand.

    Let us ask ourselves the following basic question: if Polkinghorne and his ilk confined themselves to suggesting that the physical world provided evidence for the existence of a “Creator”, would we be paying any attention?

    My own answer is a resounding “no”. (If god is the name we want to attach to our scientific ignorance, I am fine with it, provided scientists are free to push the boundaries of knowledge unimpeded by a prescriptive creator.)

    The critical step is the “ergo Jesus” fallacy. Polkinghorne and his Templeton-funded accomplices cynically use their scientific credentials as a smokescreen to provide a respectable justification for they really want to achieve, namely the public endorsement and acceptance of a very specific (and revealed!) kind of morality.

    • Posted February 23, 2012 at 6:04 am | Permalink

      With respect, _not_ giving attention to the Polkinghornes of the world – will let irrationality fester.

      We all have a duty to stamp out wilful lying, irrationality, and incompetence whenever we find it if we want to help ensure a society where truth is preferred to lies and well-being preferred to harm.

      • Manfredi La Manna
        Posted February 23, 2012 at 6:24 am | Permalink

        With even greater respect, I am afraid I did not explain myself clearly: I am all in favour (who would not?) of “stamping out wilful lying, irrationality, and incompetence”, but as an economist I have to believe in the optimal allocation of scarce resources. If you agree that the “ergo Jesus” is the crux of the matter, why waste time on a smokescreen?

        • Posted February 23, 2012 at 8:20 am | Permalink

          The “ergo Jesus” part is irrelevant to me. The problem for me is that there are missing steps between the 2 premises and the “ergo Jesus” conclusion.

          The crux for me is the faulty (by omission) reasoning, not the conclusion.

        • Posted February 24, 2012 at 8:43 am | Permalink

          Well, because helping people to escape silliness of thinking is worthwhile even if the effort seems disproportionate in terms of some random Internet person’s particular idea of optimal allocation of scarce resources.

          A valuable thing about criticising arguments is that we can ignore the idea of who is making the argument. That is, we can break the idea some of these people have that their qualifications and reputation guarantee respect for the stupid things they say.

          • Manfredi La Manna
            Posted February 24, 2012 at 2:40 pm | Permalink

            As a “random Internet person” I was merely trying to point out that:
            (i) as a variant of Occam’s razor, perhaps it is not wasteful to highlight the weakest point in an argument;
            (ii) I do not find intellectually satisfying criticizing a (weak) line of reasoning on the basis of its lack of logic/coherence/etc., when it is the “ergo Jesus” (unwarranted) conclusion that is doing real damage to actual people today.

            Yes, I do stick to my claim that intellectual and practical battles should be fought efficiently.

  4. AndreSchuiteman
    Posted February 23, 2012 at 6:01 am | Permalink

    In the evenings these days (to the detriment of my sleep), I’m reading Polkinghorne and Alvin Plantinga to learn about the brand of “sophisticated” theology that tries to reconcile faith and science.

    I would think reading this stuff after a long day’s work is more effective than sleeping tablets.

    But to address the issue at hand: is a universe that no branch of mathematics can describe even logically possible? A universe in which the Central Limit Theorem would not hold? A universe in which 1 + 1 2?

    And if it was, why would the existence of our universe be explained by positing the prior existence of something that is even less comprehensible? God of the Gaps meet the Anthropic Principle!

    • AndreSchuiteman
      Posted February 23, 2012 at 6:04 am | Permalink

      1 + 1 not equal to 2

      • Maverick
        Posted February 23, 2012 at 8:11 am | Permalink

        In a non-conservative universe, you probably would have 1+1\=\2. However, I don’t know if non-conservative universes can exist. That said, if we had developed in such a universe our brains, logic, and math would have developed to fit it, so it probably would appear logical to us.

        • Posted February 23, 2012 at 8:25 am | Permalink

          If Sally and Bob each have one apple, and they each give their apples to Marie, who previously had no apples, Marie now has two apples.

          If the result of that transaction is anything other than Marie having two apples and Sally and Bob having no apples, then conservation is violated.

          If you put even a bit of thought into the matter, you can quickly realize that anything but conservation is unstable.

          If the final result is nobody with any apples, the universe will be devoid of apples before anybody would have a chance to know what an apple is.

          If Marie winds up with more than two apples, or if Marie gets her two apples plus either Bob or Sally still have an apple, then the universe will soon be overrun with apples.

          If, even after the exchange, Marie has no apples and Sally and Bob each have one apple, then no change is possible and the universe is stagnant, forever sitting there doing nothing.

          I personally define “magic” as “anything which violates conservation.” I actually have a lot of fun examining various stories for where the conservation-violating hole is. In the case of religions, of course, it’s never particularly hard to find….

          Cheers,

          b&

          • Steve
            Posted February 23, 2012 at 8:33 am | Permalink

            Excellent.

          • dale
            Posted February 23, 2012 at 9:44 am | Permalink

            If two drops of rain starts falling and as they fall they merge with one another, how many drops of rain do you end up with?

            • Posted February 23, 2012 at 10:20 am | Permalink

              Don’t be silly.

              If the first drop had a volume of 7 mm³ and the second drop had a volume of 9 mm³, then the result is a single drop of 16mm&sup3…which is too large for a stable raindrop in our atmosphere and is likely to split again into another pair of drops, each of roughly 8mm³ volume. And, in all cases, there will be additional complications from splashing, evaporation, condensation, and the like.

              You may think you’re being clever by making bad puns based on our language’s ambiguous usage of singular nouns to refer to mass quantities, but all you’re doing is being pointlessly obfuscatory.

              Now, if your super-drop had more or less mass / energy than the combination of the two drops, it’d be a different story. But, as it is, you’re just not a very cunning linguist, I’m afraid.

              Cheers,

              b&

              • Dan L.
                Posted February 23, 2012 at 11:13 am | Permalink

                You seem to be completely ignoring everything but the field of reals. Why do you rule out the possibility of universes better described by other algebraic systems?

                In abstract algebra, there are two four-element groups. Addition works differently under these two groups, e.g. 2+3=1 in one and 2+3=4 in the other (IIRC). If you look at groups with infinite elements you have even more possibilities for defining addition. I can’t imagine a world in which addition worked differently but that might be saying more about my imagination that it does about possible worlds.

              • Posted February 23, 2012 at 11:32 am | Permalink

                Why do you rule out the possibility of universes better described by other algebraic systems?

                The universe we have is already very well described (in certain circumstances) by these other algebraic systems.

                But the reals — or, more specifically, counting numbers — are what one uses when one wishes to count unitary things. And counting unitary things is exactly what conservation is all about.

                If you want to work with imaginary numbers, then, by all means, become an electrical engineer. But you’d be an idiot to try to balance your checkbook by attempting to take the root of a negative number, just as you’d be a fool to use a hammer to make a clean cut in a two-by-four.

                Cheers,

                b&

              • Dan L.
                Posted February 23, 2012 at 11:55 am | Permalink

                But the reals — or, more specifically, counting numbers — are what one uses when one wishes to count unitary things. And counting unitary things is exactly what conservation is all about.

                That’s true in our universe. I’m trying to suggest that there may be other possibilities. You don’t have to buy it but pointing out how counting works in our universe doesn’t tell me anything about what’s possible in other universes.

                If you want to work with imaginary numbers, then, by all means, become an electrical engineer. But you’d be an idiot to try to balance your checkbook by attempting to take the root of a negative number, just as you’d be a fool to use a hammer to make a clean cut in a two-by-four.

                Umm, the complex numbers include the reals as a subset. So technically I do balance my checkbook with complex numbers. In particular, addition is defined identically on both R and C. I’m talking about algebraic systems in which addition actually works differently.

              • Yiam Cross
                Posted February 23, 2012 at 4:10 pm | Permalink

                F*&$ng scientists, no sense of humour.

              • Another Matt
                Posted February 23, 2012 at 4:18 pm | Permalink

                You may think you’re being clever by making bad puns based on our language’s ambiguous usage of singular nouns to refer to mass quantities, but all you’re doing is being pointlessly obfuscatory.

                And even this can be teased out in our language pretty easily – do you say “less than seven drops of water” or “fewer than seven drops of water”? Context is the only way to find out, and then the two concepts (individual drops vs. the mass quantity) are totally separate.

              • Posted February 23, 2012 at 6:20 pm | Permalink

                Um, Matt… 

                Unfortunately, at least in British English, “less” seems increasingly to be doing the work of “fewer” (and “amount” the work of “number”). Even among university students; it’s not uncommon to hear, “Less people went this year” (or “The amount of people …”)

                Supermarket checkout signs saying “Five items or less” are ubiquitous…

                /@

              • Another Matt
                Posted February 23, 2012 at 6:48 pm | Permalink

                Ant,

                Would this mean someone is wrong somewhere other than on the internet? I’m sorry, my beliefs do not allow for this to be true.

              • Kharamatha
                Posted February 24, 2012 at 8:37 am | Permalink

                Fewers are lessers. All fewers are lesser amounts, and at least some lessers are fewer instances.

          • David Evans
            Posted February 24, 2012 at 3:48 am | Permalink

            That’s a lovely argument, but it overstates the case. Suppose after the transaction Sally and Bob have no apples, but Marie can have 1, 2 or 3 apples with an appropriate probability distribution, symmetrical about 2. Then the number of apples in the universe would be conserved on average, though the participants might be rather confused.

            • Kharamatha
              Posted February 24, 2012 at 8:38 am | Permalink

              It would only remain conserved until a min-maxer or save-scummer were born.

          • IW
            Posted February 24, 2012 at 6:07 am | Permalink

            Does this business with Apple apply to iPads as well as iPhones?

            • Posted February 24, 2012 at 6:25 am | Permalink

              Yep! Hold down e to see and shoose among ę, ē, ė, ë, é, è, and ê, for example.

              Plus, 0 also gives you °; s, ß; &, §; ?, ¿; and so on.

              Plus you can set it to let you switch to alternative alphabets, such as Γρεεκ.

              /@

              #applefanboy

              • Posted February 24, 2012 at 6:26 am | Permalink

                Of course, it doesn’t always help you type accurately…

                *choose

                (Where is autocorrect when you actually need it? ;-) )

              • Posted February 24, 2012 at 6:29 am | Permalink

                Also… I might have misunderstood the context of your question…

                :-X

        • Posted February 24, 2012 at 4:10 pm | Permalink

          But we already know that addition is very different from various *factual* relations, for example that of juxtaposition. One water drop on my window juxtaposed with another is not two drops, bu one drop (larger, usually).

          There are formal studies of parts and wholes (mereology) and stuff like that deal with the “naively pure” mathematics which is really nothing f the kind.

    • John K.
      Posted February 23, 2012 at 6:46 am | Permalink

      In binary, 1 + 1 = 10

      • AndreSchuiteman
        Posted February 23, 2012 at 7:04 am | Permalink

        Are you suggesting that one plus one does not equal two in binary?

        • Steve
          Posted February 23, 2012 at 7:14 am | Permalink

          Andre, forget not that 10 is 2 in binary.

          • AndreSchuiteman
            Posted February 23, 2012 at 7:34 am | Permalink

            Steve, I’m not that stupid.

          • Posted February 23, 2012 at 9:07 am | Permalink

            No… 10 is 1010 in binary.

            I think you meant to say, 10 in binary is 2 in base 10 (any base > 2, actually). ;-)

            /@

            • AndreSchuiteman
              Posted February 23, 2012 at 9:50 am | Permalink

              Can we all fix our base first, please?

              • sasqwatch
                Posted February 23, 2012 at 10:25 am | Permalink

                all your bases are belong to us

              • IW
                Posted February 24, 2012 at 6:11 am | Permalink

                Who’s on first?!

              • Posted February 24, 2012 at 6:13 am | Permalink

                Yes.

              • Posted February 24, 2012 at 8:09 am | Permalink

                According to Roger, you have to ask Ouvre, Unger & Dunn.

                Cheers,

                b&

              • Kharamatha
                Posted February 24, 2012 at 8:40 am | Permalink

                Enough with this baseless speculation.

        • dale
          Posted February 23, 2012 at 9:47 am | Permalink

          If it is 10 o’clock now and we meet in 5 hours what time do we meet using a 12 hour clock?

          10 + 5 = 3.

          There are many more mathematical structures beyond the simple real numbers.

          • Another Matt
            Posted February 23, 2012 at 12:57 pm | Permalink

            Nope. 10 + 5 is congruent to 3 mod 12. The “equals” sign isn’t used properly here — you need the “congruence” sign instead.

            • Dan L.
              Posted February 23, 2012 at 1:14 pm | Permalink

              The actual mark you use to represent an equivalence relation like this is arbitrary. In abstract algebra you typically just use “=”, sometimes with a subscript, exactly as dale did. The twelve element cyclic group isn’t a great example of a group whose sum is defined differently than Peano addition on the naturals but I think dale has the right idea here actually.

              • Another Matt
                Posted February 23, 2012 at 1:28 pm | Permalink

                OK, fair point, and I’ll be happy to defer to a mathematician on any of this.

                But I think we’re both saying that the concept “plus” and “equals” in these two cases (naturals vs. mod 12) are analogies, not “the same concept.” For one thing, addition mod 12 does not have the same relations to, say, multiplication and division mod 12 (division mod 12 is not even defined) as it does to addition on the naturals. When you invent a new definition, new things fall out logically, and then it’s largely a matter of seeing how far the isomorphisms between systems go, and when they aren’t properly isomorphic it’s useful even to show “analogous behavior.”

                Inventing definitions is required even just to make observations in the first place.

              • Dan L.
                Posted February 23, 2012 at 1:45 pm | Permalink

                Well, to be clear, I’m not a mathematician. I have an undergraduate mathematics degree and this abstract algebra stuff is only half-remembered for me since I don’t use it regularly. If a working mathematician commented with a good reason to think there aren’t any internally consistent sums on infinite groups besides Peano addition I’d certainly listen (and ask a few followup questions).

                Actually, addition on the cyclic group mentioned by dale is the same as Peano addition. You’re right that he’s doing something sneaky with the “=”. The more correct terminology is to say +== where is an equivalence class defined by = iff X mod 12 = Y. So you pretty much nailed it.

                But I’m fairly certain there are other groups where “+” is not Peano addition but where “=” is plain old equality. I think dale was just trying to point out there are a lot of possible algebraic systems out there and the only reason we use Peano addition on the naturals is that it’s what seems to work. Agree 100% with what you say about definitions.

              • Dan L.
                Posted February 23, 2012 at 1:51 pm | Permalink

                Crap, I tried to use the traditional angle brackets in the equations which obviously didn’t translate to HTML that well. Let’s try that again:

                The correct terminology is:
                [10] + [5] = [15] = [3]
                where [X] = [Y] iff X mod 12 = Y

                So yeah, once again, sneaky redefinition of “=”.

              • Another Matt
                Posted February 23, 2012 at 2:44 pm | Permalink

                I’ll totally agree that there are many other totally consistent algebraic structures that could find use somewhere. My favorite one involves:

                http://en.wikipedia.org/wiki/Real_projective_line

                Dale’s example is a good one because modular arithmetic is useful for modeling behavior that seems cyclic — we need a different algebra to do math with cyclic time.

                It’s worth noting that the addition group mod 12 contains all of the mod 12 integers, but the multiplicative group in mod 12 only has 4 members (1,5,7,11). And the usual gradeschool definition of multiplication (you add this number to itself this number of times) even works to define the operation algorithmically.

                But unless the base is prime there’s no guarantee that multiplication will have an inverse for non-zero integers — 1*2 [=] 7*2 mod12. This is just as well – when we’re using modular arithmetic to model cyclic time, for instance, it’s clear what “10 hours from 10:00″ means, but not “twice 10:00″ or “half of 2:00.” It would be interesting to see what would have happened had we used an 11 or 13 (or 23) hour clock, where all non-zero multiplication has an inverse; maybe we would have had made up ideas like “3:00 times 5:00 is 2:00″ (mod 13). Such ideas do function in music theory which is also traditionally a mod-12 system.

              • Dan L.
                Posted February 23, 2012 at 3:07 pm | Permalink

                Notice that your multiplicative group mod 12 is isomorphic to the 4-element cyclic group under normal addition. This is actually a pretty good example of how you can redefine what “+” means and still come up with a perfectly consistent, coherent algebraic system. Interesting idea about the even-tempered scale being a group, I’ll have to do some research/thinking about that.

              • Another Matt
                Posted February 23, 2012 at 3:22 pm | Permalink

                It’s not quite the C4 group – I think it’s the C2XC2 group (isomorphic to the Klein four-group). This is because 5,7, and 11 are their own inverse — there’s no way to get a multiplicative cycle through all four elements.

                However, the multiplicative group for mod 13 is isomorphic to the additive group for mod 12 (C12), and that would prove your point nicely — but even such “redefinition” is still a little sketchy when the isomorphism is between systems based on concepts are way different cognitively.

                If you want more info on group structures in music, the best place to start is “Generalized Musical Intervals and Transformations” by David Lewin.

              • Yiam Cross
                Posted February 23, 2012 at 4:22 pm | Permalink

                I think this is why the rest of the world hates mathematicians.

                The bottom line, as I see it and I’m no mathematician, is that you all appear to be talking about different ways of representing the same thing. 5 hours have passed whether you represent them with a 12 hour clock, a 24 hour clock, a decimal clock or an egg timer.

                The important question, and the one I think you’re all missing, is, if a tree falls in the forest and there’s no one there to record what time it fell, will one hand clapping be enough?

        • John K.
          Posted February 23, 2012 at 1:22 pm | Permalink

          I was mostly being snarky, but the “truth” of a statement like “1 + 1 = 2″ is really just a widely agreed upon axiom, including the base 10 Arabic numeral system and the definition of addition.

          And to be picky, in binary “one plus one equals one-zero” not “one plus one equals two”.

          • AndreSchuiteman
            Posted February 23, 2012 at 2:24 pm | Permalink

            Actually, in Russell & Whitehead’s Principia Mathematica, 1 + 1 is a theorem, not an axiom. If I remember correctly, it took them a few hundred pages to prove it.

            • AndreSchuiteman
              Posted February 23, 2012 at 11:51 pm | Permalink

              1 + 1 = 2

            • Posted February 24, 2012 at 4:13 pm | Permalink

              As I’ve mentioned previously, theoremhood (and axiomhood) are relative to a system. So, yes, in Principia Mathematica, 1+1=2 is a theorem. It might be an axiom in some other system. And is, trivially, since you can easily create one where it is.

  5. Jeffrey Shallit
    Posted February 23, 2012 at 6:03 am | Permalink

    I’m not impressed by the “unreasonable effectiveness of mathematics” argument, for two reasons:

    1. We’re trying to understand the world. Tools that work we use, and tools that don’t we discard. It is hardly possible to maintain that it is “unreasonable” that the tools we settled on are effective, when the others have been discarded. Perhaps, in an alternate universe, dancing and architecture would be used to understand the universe, and Wigner-prime would be prattling about how unreasonably effective they are.

    2. Mathematics is great for some things, but it is hardly “unreasonably effective” even in physics. Consider the following scenario: at time t I release a single atom of tritium in my office. Now I ask, where will it be, exactly in time t + 1 hour? It’s not like physics is capable of telling me with any accuracy.

    • TJR
      Posted February 23, 2012 at 6:42 am | Permalink

      I once tried to explain elementary ballistics through the medium of dance.

      It didn’t work.

      • Steve
        Posted February 23, 2012 at 7:04 am | Permalink

        What was that, a kind of shot in the dark?

        • TJR
          Posted February 23, 2012 at 7:28 am | Permalink

          Dancing in the dark?

      • Yiam Cross
        Posted February 23, 2012 at 4:05 pm | Permalink

        Went down like a lead balloon?

    • clive photo
      Posted February 23, 2012 at 4:04 pm | Permalink

      God knows. I suppose.

  6. Jeff Engel
    Posted February 23, 2012 at 6:09 am | Permalink

    I think this may be, in part, another instance of Douglas Adam’s sentient puddle being amazed at how well the dip is designed to fit it.

    We don’t bring mathematics initially to the universe and find ourselves amazed that it can usefully model things in it. We get mathematics, at the beginning, from the universe, through individual learning, successful memes, and from dispositions inherited from ancestors who wouldn’t have been ancestors had the universe not cooperated with their burgeoning mathematical models.

    • Steve
      Posted February 23, 2012 at 6:14 am | Permalink

      You beat me to this point, Jeff.

    • Ken Browning
      Posted February 23, 2012 at 10:28 am | Permalink

      And by extension, it’s a god-of-the-gap and therefore an Intelligent Design argument. Polkinghorne marvels at the beauty and complexity of mathematics and makes the leap that “it” had to be created by an eternal personality.

      How can a Personality exist if there is no concurrent and already existing mathematics?

      • Posted February 24, 2012 at 4:22 pm | Permalink

        The funny thing is one does get theistic disagreement on this, too.

        Descartes thought mathematics was created by god, and that he could make arithmetic “work differently”. To contrast (and there are other positions) it seems that Plato thought that god and the forms, including mathematical ones, were independently existent, etc.

  7. Steve
    Posted February 23, 2012 at 6:09 am | Permalink

    In what way was it determined the level of unreasonableness of mathematics to be effective?

    Does this not assume facts not in evidence?

  8. 1000 Needles
    Posted February 23, 2012 at 6:10 am | Permalink

    Zach Weiner seems to be channeling you in today’s comic: http://www.smbc-comics.com/index.php?db=comics&id=2529#comic

    If God could make a universe that obeyed a simpler set of rules, why wouldn’t it do that? Unnecessary complexity makes no sense if “God dun it.”

    However, unnecessary complexity does make sense if math is our clumsy, but best, attempt at describing the workings of the universe. Surely, if the universe operated as elegantly as Polkinghorne supposes it does, we’d already have a Grand Unified Theory.

    • Yiam Cross
      Posted February 23, 2012 at 4:25 pm | Permalink

      And we’d each be given a copy when we entered the world, along with some proper training in how to use it. And safety goggle, I’d hope.

  9. GBJames
    Posted February 23, 2012 at 6:10 am | Permalink

    I am not physics-savvy, although I did very much enjoy Lawrence Krauss’ A Universe from Nothing. And I certainly am not math-savvy. (although The Number Devil by Hans Magnus Enzensberger was great fun to read with my kids.)

    Still, does one need to be highly trained in math and physics to recognize the old god-of-the-gaps? Polkinghorne is amazed by the universe and has questions that he has no answer for. Big deal. He’s just wrapping a lot of fancy vocabulary around the tired old arguments.

  10. Posted February 23, 2012 at 6:11 am | Permalink

    Is it possible, d’ya really think, that Polkinghorne does not recognize that presently unanswered questions like “Why X?” or “Howcome Y?” or “From Whence Z?” do not constitute evidence that “Goddidit” is the surely correct answer? Unanswered questions are simple evidence that not all question are answered, which NOBODY DENIES — except evidently Polkinghorne, who I suppose truly does think “Goddidit” is the surely correct answer to any otherwise unanswered question (that, or else he knows better but thinks everyone else will stupidly think so and thus is willing to MISlead people to [alleged] God). I keep trying to think better of Polkinghorne, but he keeps making that SO hard to do…

    • Scott near Berkeley
      Posted February 23, 2012 at 11:17 pm | Permalink

      “Why” questions (e.g. “Religion is about why it is so.”) are essentially meaningless. They carry no value. If a “Why…” question was legitimate in the eyes of a Deity, that Deity would have no other work but to accumulate information: the answers to all the general =and= all the specific ‘why’ questions. The answer to one “why” questions simply generates, exponentially, more “why” questions. Since we can exist, and never ask “why”, certainly, there is no Deity an infinite amount of responses to every why. Ergo, no Deity is in existence that is concerned with “Why….” anything!

  11. Ray
    Posted February 23, 2012 at 6:13 am | Permalink

    The God hypothesis is worse than useless in solving the regularity problem. You say the universe is that way because God wanted it that way, but then you’re left with the problem of why God wanted it that way. And the only way you can answer that is by assuming some sort of regularities in God’s thought process. So in the end you’re stuck with explaining not just any regularities, but regularities of the sort that have repeatedly been shown to be better explained in terms of the regularities we were trying to explain in the first place.

    • Havok
      Posted February 23, 2012 at 4:40 pm | Permalink

      I think this is an important point to try to get across – there doesn’t seem to be any way to get “the universe as we find it” from the God of bare theism without introducing a boat load of other assumptions. In the end I think you’d end up with an even bigger “question” to be answered than you began with.

      Dawes in Theism and Explanation makes a very similar point, I believe.

  12. bloodyhell
    Posted February 23, 2012 at 6:19 am | Permalink

    One problem is that mathematics, although amazingly effective in some branches of science (eg. physics), really isn’t that effective in a larger context. The use of mathematics in biology, for example, is typically ad-hoc and generally unsuccessful on the large scale. Why does God play favorite with physicists? In fact, many branches of physics are having as much trouble with math as biology. Anything that involves many interactive elements seem to defy elegant mathematical formulation. If you concentrate on the part of the nature world that can be handled well by math and ignore everything else, of course math seems to be amazingly effective.

    The situation is similar to the fine-tuning argument. The universe is supposedly carefully tuned… but only if you ignore the many features of the universe that is outright hostile to us. So the theists are left with a position where the universe is fine-tuned but also buggy. A unknowable will of God must be evoked to answer the question why God is so careful in this aspect of the universe but not the others. Similarly, somehow God makes some features of the universe so mathematical but not the other parts. Why? Because God works in mysterious ways.

    • Yiam Cross
      Posted February 23, 2012 at 4:32 pm | Permalink

      “Why does God play favorite with physicists? ”
      Because god, in his infinite mercy, does not torture the afflicted. As any fule kno (for those fortunate enough to have been brought up on 1066 and all that)

      Or, to put the whole thing in it’s proper place
      “‘Reality,’ sa molesworth 2, ‘is so unspeakably sordid it make me shudder.'” (Whizz for Atomms)

    • Posted February 24, 2012 at 4:27 pm | Permalink

      Some would say that’s simply lack of creativity of the relevant sort. Personally, I think it is a great difficulty in doing qualitative exact theories in novel domains, though there are of course quantiative models in biology already as well.

      And, crucially, a slightly imprecisely stated theory in natural language which is nevertheless testable, etc. and even something like true is much more useful than an exact pseudobiology, certainly. For example, Darwin’s work is not exact in the technical sense I allude to, and Dembski’s nonsense is, but it is exactly that – it is nonsense (or almost completely false).

  13. Posted February 23, 2012 at 6:21 am | Permalink

    “Before I address your argument – is it intended strictly to reassure those who profess your belief but have doubts, or is it actually intended for anyone else at all to hear it and thus respond to it?”

    • GBJames
      Posted February 23, 2012 at 6:42 am | Permalink

      Say what? Are you asking a question or quoting someone else’s question out of context?

      • Posted February 23, 2012 at 6:54 am | Permalink

        It’s my reaction to the completely shit arguments theologians put forward.

        • GBJames
          Posted February 23, 2012 at 7:09 am | Permalink

          Ah. Got it.

  14. Posted February 23, 2012 at 6:24 am | Permalink

    If the universe cannot have self-contradictory properties, it must follow mathematical laws. Math is an extension of logic.

  15. Daryl
    Posted February 23, 2012 at 6:25 am | Permalink

    My response to these arguments is mild annoyance. God is an answer, but it’s not an explanation. These arguments only have weight to someone who already believes in a god.

    I don’t have the background to give the perspectives you’re looking for, so I’ll shut up now and look forward to reading responses from the physics and math-savvy readers.

  16. Posted February 23, 2012 at 6:26 am | Permalink

    Polkinghorne is suggesting a theory in which the universe might have been chaotic and incomprehensible. Further, his theory posits a personal deity whose conscious attention is required to keep things from falling apart.

    First, this is no different from ancient, long since dismissed ideas about celestial mechanics. To hold to such today is as patheitcally laughable as insisting on the validity of astrology.

    Second, it is a skyhook such as what Richard describes with respect to Evolution. If Jesus is what keeps the stars from falling from the sky, then what super-god is it that keeps Jesus’s guts from falling out of his abdomen? If no such super-god is needed to explain Jesus’s intestinal fortitude, why is Jesus needed to explain why the sky isn’t falling on Chicken Little?

    On the other hand, if one simply observes that complex phenomena can only arise in the presence of simpler supporting phenomena, the answer becomes obvious: explain the simple, and you’ve also explained the complex.

    And, sure engouh, that’s exactly what Hawking has done of late. We observe quantum fluctuations that “just happen” which create virtual particles with a net mass / energy of zero, and it would seem that the Big Bang itself is but a variation on that same theme. Rather obvious, really, in hindsight, once you think about it.

    It also goes without saying that Polkinghorne really first must demonstrate that the sort of anarchy he imagines is even a coherent concept in the first place, and he’s not even pretended to attempt to do so. I can invent all sorts of superheroes to save me from any manner of horrific indescribable fates, too. Doesn’t mean any of it has any bearing on even theoretical reality.

    Cheers,

    b&

    • Posted February 23, 2012 at 7:52 am | Permalink

      Way to work in the intenstine reference :P

    • Posted February 24, 2012 at 4:33 pm | Permalink

      There’s a tradition someone started claiming god is needed because without the “sustaining” nature of ths thing the (non-god) universe would lapse from existence. As if that is consistent with what we know about basic conservation laws (even if they are hard in GR).

      Worse still for the believer, because that makes god concurrent with every last event in the (non-god) universe, god must thereby agree that it is thereby worth it. This creates a massive problem of evil, etc. Leibniz admittedly took this horn of the dilemma, but as a matter of course, most theists are unconfortable with his answer.

      • Moochava
        Posted February 24, 2012 at 5:50 pm | Permalink

        This is consistent with my understanding of video game magic, where once you kill the evil wizard, his curses evaporate, his summoned monsters vanish, and the flying castle you were fighting him in crumbles. I believe it’s called the Rule of No Ontological Inertia.

        Related note: religious people HATE it when you point out that their reasoning works really well in video games.

  17. Sigmund
    Posted February 23, 2012 at 6:29 am | Permalink

    People were amazed by the idea that the earth went around the sun rather than the other way around. It was ‘unreasonable’ that the earth orbits the sun – based on the evidence that people possessed at that time.
    Humans are pattern forming animals but we tend to deduce patterns from a combination of innate propensities (such as fear of predators or height) and learned experiences. A novel means to deduce more accurate models and patterns that does not rely on these former abilities is bound to seem ‘unreasonable’ – in the sense that they are often counterintuitive to the way we view problems. Mathematics, logic and the scientific method seem “unreasonably effective” to us because human are not logical animals. We are pretty close but are pretty easily tricked by expectations, bias and a lack of appreciation for experimental statistics.

    • Scott near Berkeley
      Posted February 23, 2012 at 11:22 pm | Permalink

      Nicely put.

      If one studies humans in terms of “what made this hunter successful?” you learn a lot of insights.

  18. Chris
    Posted February 23, 2012 at 6:29 am | Permalink

    Physics may postulate the *possibility* of a multiverse but it so happens that Christians *require* a multiverse model for their god.

    I realise this will come as desparate news to, well, basically all of them, but an omniscient god who is omnipotent enough to be able to change his mind or answer prayers requires an infinite number of potential universes to operate in. While it might not be necessary for multiple universes to exist simultaneously, their god’s omniscience means that the current universe only exists until god changes what was going to happen anyway and we all go careering off down a different path.

    • Steve
      Posted February 23, 2012 at 6:48 am | Permalink

      Or not… what’s to say this couldn’t happen in one universe?

      • Chris
        Posted February 23, 2012 at 11:52 am | Permalink

        Omniscience. God can see everything that will happen until the end of time. That’s a universe mapped out right there. Until someone sends another prayer up and it all has to be chnaged again.

        • Posted February 23, 2012 at 11:57 am | Permalink

          All the omni-properties are incoherent. An omniscient entity cannot know what ignorance is like. An omnipotent entity is incapable of suicide, for it thereby becomes powerless to effect anything after death. And so on.

          Speculating about the logical consequences of omni-mumble deities is as useful as speculating about the Kryptonite content of Superman’s condoms. Whether or not either is entertaining I leave for the reader to decide.

          Cheers,

          b&

  19. Linda Grilli Calhoun
    Posted February 23, 2012 at 6:31 am | Permalink

    “We don’t understand why there are physical laws that behave with regularity, but that lack of understanding doesn’t point to the existence of a creator—much less Polkinghorne’s Christian creator who birthed Jebus.”

    My immediate reaction to this article was that it is a “god of the gaps” argument.

    Just because we may not understand at this point in time doesn’t mean that we won’t at some point in the future.

    Also, mathematical understanding, for me, points away from a god, not toward. The description of events as “miracles” posited by believers as “evidence” just leads me to conclude that these events are statistical outliers, fully explained by randomness. Spontaneous remission of cancers is a perfect example of this. Remissions happen in believers and skeptics with roughly equal frequency. Statistics rationally explains that; religion does not. L

  20. Paulo Jabardo
    Posted February 23, 2012 at 6:32 am | Permalink

    Perhaps asking why the success of mathematics is so great in describing the physical world is almost a tautology! Up until 200 years ago every scientist was mathematician and every mathematician was a scientist. Mathematics was a tool developed to describe an ordered nature. No wonder it does it well!

    On the other hand if you do experimental work, math does not work so well. It requires lots of work and imagination to get it to fit the data. Just think about gravitation. Everyone knows that all objects when free falling undergo the same acceleration. Try doing it yourself. Not only do objects not fall with the same acceleration but some even “fall” laterally (think of a leaf wobbling down). It took centuries of improvement in measurement techniques and gravitational hypothesis until we could understand how things fall and be able to recognise aerodynamic effects. But until then free fall was almost a random event and today we know that all objects do not free fall with the same acceleration: it depends on the gravitational field and even that is wrong: people still haven’t found Vulcan orbiting the sun closer than Mercury.

    A theist will probably say that it requires faith to believe that there is order. I say that if there was no order in the universe there really wouldn’t be much point in doing science: it would amount to “stamp collection”. You shouldn’t even bother to do statistical analysis of your stamp collection because if this turned out something interesting (other than nothing is correlated to anything else) this would imply order – quantum effects are random but the wave functions, related to probability densities, can be integrated. Turbulent flows are random but exhibit structure.

    Now, if there were no order in the universe, the only way for us to have this discussion is for a god to be pulling the strings, otherwise hens could be laying gold eggs and the next time I take a dump I should watch carefully the toilet so I don’t drown my possible new born baby crocoduck.

    • Steve
      Posted February 23, 2012 at 6:55 am | Permalink

      people still haven’t found Vulcan orbiting the sun closer than Mercury.

      WTF? Maybe this Vulcan you speak of is where JFC’s bones are buried.

      • Paulo Jabardo
        Posted February 23, 2012 at 7:29 am | Permalink

        Astronomers observed irregularities in the orbit of Mercury and, based on the discovery of the outer planets of the solar system, calculated the position of an inner planet and even named it: Vulcan. It turned out to be a relativistic effect of the orbit of mercury.

  21. Occam
    Posted February 23, 2012 at 6:35 am | Permalink

    I can’t resist quoting one rejoinder:

    “… if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide.”
    In very loose terms: brain maps universe.
    Guess who wrote this?
    I’ll save you Googling: Holy Dad. Yes, Ben XVI Ratzinger. Here.
    Sure enough, he blows it, too, further on in that paragraph. Just not quite as blatantly as Polkinghorne. Ratz tries to sell a tortuous gnoseological argument, whereas Polkinghorne merely jumps for faith, abandoning reason and evidence in mid-sentence like just so many dirty diapers:

    Theology, however, can step into the breach. Science has disclosed to us a world which, in its rational transparency and beauty, is shot through with signs of mind, and religious belief suggests that it is indeed the Mind of the Creator that lies behind the wonderful order of the universe. (p. 73)

    “shot through with signs of mind”: yes, our own. Wonderful or otherwise, the only order of the universe we can discover is the one that our nervous system is attuned to.
    More precisely: has evolved being attuned to.

    • Occam
      Posted February 23, 2012 at 6:39 am | Permalink

      Oops, indentation failure: my nervous system is not sufficiently attuned to ‘blockquote’.

  22. TJR
    Posted February 23, 2012 at 6:38 am | Permalink

    Agree with most of the comments above, especially #2 and #5.

    Mathematics is used to model the universe, but even in physics Newtonian mechanics and General relativity model gravity in quite different ways.

    Its not remotely surprising that maths has provided massively useful models in some areas (physics) but not in others (economics).

    I’ll quote it again as it is always worth remembering:

    “All models are wrong, but some models are useful”.

    As we all suspected, Sophisticated Theology is a quantum effect so that when you observe it the wavefunction collapses and it disappears.

    • HaggisForBrains
      Posted February 23, 2012 at 11:16 am | Permalink

      As we all suspected, Sophisticated Theology is a quantum effect so that when you observe it the wavefunction collapses and it disappears.

      +1

      • Posted February 23, 2012 at 11:26 am | Permalink

        [It] had softly and suddenly vanished away…
        For the Snark was a Boojum, you see!

        /@

  23. Steve Smith
    Posted February 23, 2012 at 6:40 am | Permalink

    The universe has not only proved to be astonishingly rationally beautiful, affording scientists the reward of wonder for all the labours of their research. Why are we so lucky?

    Polkinghorne’s “mathematical” argument is no better than the blatantly bogus account of Euler’s “proof” of god that cowed the great atheist Diderot, or so goes the story: “Sir, (a+b^n)/n = x; hence God exists, answer please!”

    As for the necessity of math in physics, it’s true that we don’t know why this appears to be true, but Feynmen undercuts the logic of its necessity with this trenchant observation about the huge amount of calculation necessary to determine what happens in the tiniest amount of space-time:

    It is possible—and I’ve often made the hypothesis—that physics will not ultimately require a mathematical statement, that the machinery will ultimately be revealed—it’s just a prejudice, like one of these other prejudices.
    It always bothers me that, in spite of all this ‘local’ business, what goes on in no-matter-how-tiny-a region of space and no-matter-how-tiny a region of time, according to the laws as we understand them today—takes a computing machine an infinite number of logical operations to figure out.
    Now how could all that being going on in that tiny space?! Why should it take an infinite amount of logic to figure out what one tiny, stinky bit of space-time is going to do? So I made the hypothesis often that the laws are going to turn out to be, in the end, simple like the checkerboard, and all the complexity is from size.

    • Posted February 23, 2012 at 11:27 am | Permalink

      Hmm… Asimov told me it was e^iπ + 1 = 0.

      /@

      • Steve Smith
        Posted February 23, 2012 at 2:20 pm | Permalink

        That’s a valid argument for God’s existence (ergo Jesus), hence we atheists just ignore it.

        • Posted February 24, 2012 at 4:37 pm | Permalink

          In the technical sense of valid, there are lots of (arbitrarily many) valid arguments for god’s existence. What matters is if they are sound and persuasive, which they aren’t (or at least, they are certainly not knowingly so, without begging any number of questions).

          E.g.,

          God exists. Therefore, god exists.

          is a valid argument.

  24. Posted February 23, 2012 at 6:43 am | Permalink

    I am not a mathematician or a physicist, but as I understand it having a creator or god is actually impossible because of:
    1) Gödel’s incompleteness theorems;
    2) Quantum Indeterminacy;
    3) Poincare’s 3-Body Problem.

    Point 1) is significant as it shows that not all solutions can be determined by the very nature of mathematics. The rules of mathematics itself are incomplete, and can never be proven.

    Point 2) is significant as quantum indeterminacy prevents the level of knowledge that is needed to be able to fine-tune the universe at the point of the big bang. The ability to manipulate the proto-universe must require the ability to know the current state of that universe in order to correct any perturbations from the desired outcome. This is no more possible than being able to draw a four-sided triangle on a flat plane.

    Point 3) shows that the level of computation that would be needed in order to manipulate anything more than three particles to a sufficient degree of accuracy is impossibly large. The universe cannot be calculated and corrected in a top-down fashion that a creator would need to do. The only solution that makes sense is a bottom-up one; one that involves the anthropic principle, evolution and even multi-universes (getting the rules just right… eventually)

    Of course, I may be wrong on any of these and would very much like to hear where these impossibilities are able to be overcome by a creator god.

    • Posted February 24, 2012 at 4:39 pm | Permalink

      1) All mathematical theories strong enough to include elementary arithmetic are formally incomplete. However, one can always add the “goedel sentence” and iterate as much as you like … (This is partially why I said above that mathematics is creative.)

      2) and 3) reduce to the fact that gods violate conservation laws, etc.

  25. yesmyliege
    Posted February 23, 2012 at 6:43 am | Permalink

    Polkinghorne asserts:

    “…Science has disclosed to us a world which, in its rational transparency and beauty, is shot through with signs of mind, and religious belief suggests that it is indeed the Mind of the Creator that lies behind the wonderful order of the universe. (p. 73)…”

    If the order of the universe truly reveals the mind of the creator, then the creator suffers from a severe case of schizophrenia.

    The natural world is not orderly – it is nonlinear, chaotic, stochastic. The construct of mathematics gives us useful prognostic models of its behavior, but only up to a point. Most of mathematics assumes a linear system, but most of nature is exactly the opposite, which is why chaos theory is a brand new field.

    But chaos math only approximates the real world, and can not describe it perfectly. Ask an engineer or experimental physicist just how well mathematics describes turbulence, for example, and you will hear a fervent exasperation with Polkinghorne’s assertion.

    If “indeed the Mind of the Creator…lies behind the wonderful order of the universe” then the universe would look as ideally symmetrical as theological hyperbole would allow, and that is saying something. We would have no need for fractions, for example. Heck, every equation describing the universe would be utilize the triumvirate. Dna would be encoded with three bases not five, all solar systems would have three planets each with three moons. Insects would have three legs, not the four the Bible tells us they actually possess, and every aspect of every molecule, atom, planetary system and galaxy would be perfectly radially and trilaterally symmetrical.

    No, math is not “unreasonably effective” nor is the universe theologically orderly.

  26. sasqwatch
    Posted February 23, 2012 at 6:45 am | Permalink

    It may not be exactly what is being articulated in those passages, but it bears repeating that we don’t have Laws that somehow exist as some kind of foundation, and stuff obeys those Laws. That’s a holdover from anthropomorphizing the situation… it’s a category error from thinking of Laws as being some kind of written rules in the background. It’s putting the cart in front of the horse.

    Like said above — they, as well as math(s), are models. It (really easy maths) works the way it does because we are scaled the way we are… in-between the really tiny and the super huge. We may never get beyond a crazy-quilt of different models/methods, each applicable to different situations. This hardly seems like some overarching regularity to me. (though it’s tempting to think so, especially in physics)

    • sasqwatch
      Posted February 23, 2012 at 7:13 am | Permalink

      And of course while I was writing that, the 5 commenters above me encapsulated what I was thinking so much better. By “physics” I was meaning the easy classical stuff, or even the relatively easy idealized QED/QCD stuff. Unfortunately it’s all uphill from there.

      By coincidence, I was reading up on Godel after watching a humorous segment of Stephen Fry on QI, where he showed Godel’s formulism of an ontological proof of God. I was a bit taken aback that Godel was absolutely convinced of the existence of a personal god, despite having laid waste to the idea of some kind of completely self-consistent set of axioms for all of mathematics. Now I’m not sure how serious he was about that ontological proof.

      • Dan L.
        Posted February 23, 2012 at 11:25 am | Permalink

        In August 1970, Gödel told Oskar Morgenstern that he was “satisfied” with the proof, but Morgenstern recorded in his diary entry for 29 August 1970, that Gödel would not publish because he was afraid that others might think “that he actually believes in God, whereas he is only engaged in a logical investigation (that is, in showing that such a proof with classical assumptions (completeness, etc.) correspondingly axiomatized, is possible).”

        From the wiki on Godel’s ontological proof. It seems as though Godel really did believe in God, and the above quote if accurate in the first place seems like a bit of a joke. But what I take from it is that Godel didn’t think it proved his religious beliefs to be true, that it was an exercise in formal logic mostly unrelated to his (real) belief in God.

        • Posted February 23, 2012 at 11:28 am | Permalink

          If Godel really did believe in God, did Gödel really believe in Göd?

          :-D

          /@

          • Dan L.
            Posted February 23, 2012 at 11:51 am | Permalink

            LOL. Maybe if my job didn’t involve huge reams of HTML escapes I’d be more enthusiastic about using them in my off-time.

            • Posted February 23, 2012 at 12:35 pm | Permalink

              Or you could just type a “ö”. …

              /@

              • Dan L.
                Posted February 23, 2012 at 1:07 pm | Permalink

                What’s the keyboard shortcut for that on windows 2004 server? And Ubuntu?

              • Dan L.
                Posted February 23, 2012 at 1:10 pm | Permalink

                Never mind. The keyboard shortcuts don’t work in firefox and I’m not going to use microsoft’s shitty browser just to have the ability to memorize a bunch of useless keyboard shortcuts.

              • Posted February 23, 2012 at 1:12 pm | Permalink

                ALT-CTRL-SHIFT-M-A-C.

                That, plus a few pennies, and all you have to do is hold down the “o” key until the menu pops up listing all the accented options. Or, Edit => Special Characters if you’re looking for something more exotic.

                Really, life is waaaay too short to worry about keyboard shortcuts on primitive operating systems….

                b&

              • Posted February 23, 2012 at 1:48 pm | Permalink

                :-D

        • sasqwatch
          Posted February 23, 2012 at 5:09 pm | Permalink

          Yes, that’s my take on the situation. His “proof” was tongue-in-cheek, and probably a slam on the whole idea of a supposedly self-consistent formalism overstepping its purpose.

          My (admittedly bleak) understanding is that he differed with his friend Einstein about deism, going full-swing personal goddy-woddy on his ass.

        • Posted February 24, 2012 at 4:49 pm | Permalink

          It is slightly distorted. Gödel was a theist (though not of any particular religion) but was gravely concerned that he would be vilified for being such in this “secular age” (particularly by the logical positivists, with whom he had interacted as a young man). As for his ontological argument, as far as I can tell it was meant seriously, but not as definitive. Gödel was quite (to a fault) a perfectionist, especially about his own work, and only seems to have released (to Dana Scott, if I recall) the proof because he didn’t want it lost, not because it was necessarily conclusive. It is important to remember that although Gödel was a brilliant and accomplished man, especially towards the end of his life he was not well mentally – and as such some of his actions with regards to how he thought he was being treated are irrational and, given his death from self-imposed starvation (effectively) very harmful. A sad story, really.

  27. MichaelD
    Posted February 23, 2012 at 6:48 am | Permalink

    I think my big problem with 1. is how do you distinguish between a universe with consistent rules and behavior created by god and one where it was an intrinsic characteristic of the universe regardless of whether or not god exists. Since we have so little data on gods (no data points imo)and universe formation and design (1 data point) it seems unreasonable to extrapolate anything on this point until a fuller data set is created.

    I think my biggest problem with 2 is the kind of “unreasonable effective” part. With all the previous caveats how do you know it is or isn’t reasonable? Seems like an argument from ignorance or personal incredulity. I can’t see how this could be so thus god. Whether you can or can’t think of a way doesn’t mean god did it. You have to show that it did.

  28. Chris
    Posted February 23, 2012 at 6:54 am | Permalink

    Seems like a case of “religious folks who should know better shocked that tools developed over centuries to approximetely model the universe model ACTUALLY APPROXIMATELY MODEL THE UNIVERSE!”

    Sorry about the shouting, I was trying not to dent my forehead on the keyboard.

    • Chris
      Posted February 23, 2012 at 6:54 am | Permalink

      My spelling seems to be the only thing approximate here.

  29. Reginald Selkirk
    Posted February 23, 2012 at 6:54 am | Permalink

    The universe can be comprehended by the rational faculties of humans, and its workings appear to adhere to laws of physics.

    Wow, think about it. The universe follows natural laws, therefore God. On the other hand, if the universe didn’t follow natural laws (i.e. there were miracles), therefore God. Polkinghorne has got a winner either way.

    • Steve
      Posted February 23, 2012 at 7:02 am | Permalink

      Hence: Heads God wins, tails physicists lose.

    • Tulse
      Posted February 23, 2012 at 8:02 am | Permalink

      Yep, I think this is perhaps the best argument against Polkinghorne. In general, if a particular state and its opposite both support a hypothesis, neither state is useful for determining that hypothesis’ truth.

      • Mandrellian
        Posted February 23, 2012 at 5:11 pm | Permalink

        Or: “that which explains everything explains nothing” and so we’re back to square one: “why is this so?”

        I really have to wonder at the intellect/integrity/honesty of those who admonish others to “read sophisticated theology” and understand “nuanced belief” (or whatever other PoMo buzzwords are in vogue) before rejecting theism.

        The only thing sophisticated about sophisticated theology is the size and volume of words used – but that’s a lie anyway; they’re employed more to obfuscate and impress (and of course reinforce the existing beliefs of the theologian and justify their continuing employment/book deals) rather than actually explain, or illustrate, or even just say anything.

        I know it’s beating a dead unicorn, but in all its years of pretending to be a “way of knowing”, the only thing theology has displayed knowledge of is how to, regardless of anything anyone else says, maintain that God exists and that’s the end (and beginning) of the conversation. All theology has ever been able to demonstrate is the lengths to which its practitioners will go to not say a single valuable or novel thing, or self-examine in any meaningful or honest manner.

  30. Posted February 23, 2012 at 6:54 am | Permalink

    I’m a died-in-the-wool atheist. But the “unreasonable effectiveness of mathematics” — especially when combined with quantum mechanics’ information as a fundamental property of (subatomic) matter — does kind of give me the heebie-jeebies. I’m using the word, “information”, as it is meant in quantum mechanics: data that reveals something about the physical universe. Both seem to suggest some sort of intelligence has existed from the beginning — because that’s when the laws of physics were set (in the first second of the universe’s existence).

    Throw in the curious role of consciousness in quantum mechanics and one can’t be blamed for wondering why these things, so suggestive of “mind”, are so central to existence. Yes, I’m aware that there’s differing opinions on whether or not the role of consciousness is real or statistical but still, the jury’s still out on the matter.

    Anyway, these are things I’ve mused about and that I find fascinating in an intellectually titillating way.

    • Posted February 23, 2012 at 7:09 am | Permalink

      “Yes, I’m aware that there’s differing opinions on whether or not the role of consciousness is real or statistical but still, the jury’s still out on the matter.”

      Not among physicists, as far as I know. What’s your evidence that this is even a significant minority amongst people who can actually do the equations in 2012?

      • Posted February 23, 2012 at 7:56 pm | Permalink

        John Archibald Wheeler, Henry P. Stapp, Bruce Rosenblum, and Fred Kuttner are all contemporary physicists who endorse the von Neumann/Wigner interpretation or some variation on it.

        Of course, nobody has found an actual mind/matter connection, and it does seem unlikely, so many other interpretations have been proffered over the years and the original von Neumann interpretation has fallen out of favor.

        But that wasn’t my point. All I said was that the jury was still out. In other words, the question is still not decided.

        • Havok
          Posted February 23, 2012 at 8:10 pm | Permalink

          I thought the whole “consciousness collapses the waveform” thing had been soundly disregarded (if that it what you’re referring to), since it seems that it is interaction with the “rest of the world” which does this, not just conscious observers.

          I could be wrong about that, however :-)

        • Posted February 24, 2012 at 1:04 am | Permalink

          You haven’t answered what I’ve asked, i.e. we’d need the proportions of those who don’t too. The jury is not seriously out on this question. It really isn’t.

          • Posted February 24, 2012 at 7:19 am | Permalink

            So the matter is settled then? Is that an opinion or a fact?

        • Posted February 24, 2012 at 1:27 am | Permalink

          Yes, the jury is in, but as ever there are some who disagree with the verdict.

          If they want to appeal, then they need to present new evidence.

          /@

          • Posted February 25, 2012 at 1:03 am | Permalink

            Dodging the VERY simple question: opinion or fact?

            • Posted February 25, 2012 at 2:07 am | Permalink

              How on earth can I be dodging a question that you didn’t ask until nearly six hours later?

              You introduced the jury metaphor; I sustained that.

              So, is a jury’s verdict opinion or fact?

              /@

              • Posted February 25, 2012 at 6:06 am | Permalink

                I guess I got my answer. No, you won’t answer it, even though you’ve already replied to it.

                The phrase, “the jury is still out”, means that a decision has not been reached . . . that something is undecided. That’s all I’ve ever said. You want to make it out to be something else. You’ll have to excuse me if I don’t cooperate with you on that one.

                If your assertions are asserted as opinions, then there’s no problem. I’ve already acknowledged that literal interpretations have fallen out of favor.

                If your assertions are asserted as facts, then there’s a problem. You’ve got to get your findings published before somebody else takes your rightful place in the pantheon of great thinkers.

                The matter is NOT yet decided. Period.

              • Posted February 25, 2012 at 1:28 pm | Permalink

                Are you sure your first name is Jim, not Hector?

                /@

    • Posted February 24, 2012 at 4:51 pm | Permalink

      What role of consciousness?

      Sorry, but Deepak Chopra (or the philosophical musings of Bohr, for that matter) are not good sources for QM. The subjectivist interpretations are *provably* wrong, and have be so proved at least since 1967.

      • Posted February 25, 2012 at 1:02 am | Permalink

        Hmmmm . . . dodging the question? I never mention Chopra or Bohr. But I did mention other contemporary physicists.

        My latest (and only) question to you remains: is your position a matter of fact or opinion? Why not answer it?

        • Posted February 26, 2012 at 7:20 am | Permalink

          Read _Quantum Theory and Reality_, ed. Mario Bunge for “ghost free” QM.

          Or, later, his 1974 work in semantics also has a more rigorous treatment of the same subject.

  31. AndreSchuiteman
    Posted February 23, 2012 at 6:57 am | Permalink

    I have always found the question ‘Does π exist?’ more interesting than the question ‘Does God exist?’

  32. Divalent
    Posted February 23, 2012 at 6:59 am | Permalink

    Is there anything about the universe that religion is “unreasonably effective” in explaining? Or even merely “reasonably”?

    It seems like a new twist on the “false dichotomy” ploy: A works, so alternative B must also be true!

  33. eric
    Posted February 23, 2012 at 7:01 am | Permalink

    Not adding much new, just rehashing a lot of what people have aleady said:

    -Yeah, this is just a restatement of the anthropic principle, with all its attending flaws.

    -The term “unreasonably” seems to beg the question. If math worked better or worse, the theists would be making the same argument.

    -Like the ontological proof, even if you accept it, it seems impossible to go from “there exsits something that creates regularity” to “basic tri-omni god is this thing,” let alone “first century BC jewish man was this thing.”

    -I would propose as a radical ‘food for thought’ notion that what we think are variables and arbitrary laws may all be deductively connected to a few simple principles. IOW, the regularities of the world could be necessary in the philosophical sense, we just don’t grok how. As incomplete evidence for this hypothesis, I cite the fact that the physical sciences have continued to reduce the number of independent variables and laws over the past several centuries as deeper connections between apparently disconnected phenomena are unearthed. Empirically, there is no reason to believe this trend won’t continue.

    • Chris
      Posted February 23, 2012 at 9:16 am | Permalink

      Possibly that word ‘irrationally’ doesn’t just beg the question, but makes his argument self-refuting, since if the universe is rationally ordered, then how can mathematical modeling be ‘irrationally’ effective? It would just be rationally effective, and hence not require any ‘outside’ explanation.

      • Chris
        Posted February 23, 2012 at 10:26 am | Permalink

        Er, I meant ‘unreasonably’ – mathematics would just be reasonably effective, etc.

  34. Jim
    Posted February 23, 2012 at 7:04 am | Permalink

    I get particularly wound up by people assuming that “why” is always a valid question – particularly when “why?” is used in the sense of “for what reason?” as is clearly the case here.

    The only apparent reason for asking “why?” in this case is so that one can superficially answer it with one’s presupposed conclusion that a magic daddy in the sky exists.

    Which is really f*cking stupid.

    It seems to me that the word “sophisticated” in “sophisticated theology” would be more accurately replaced with the word “disingenuous.”

    • Darrell E
      Posted February 23, 2012 at 7:51 am | Permalink

      Very Nice.

  35. John K.
    Posted February 23, 2012 at 7:09 am | Permalink

    Mathematics is purely mental. It creates a rigorously non-contradictory and consistent system, but at its base there are all axioms that are simply agreed upon. This lets us construct powerful and precise models for things, but they are just models. Math alone does not cause anything; it only has a chance to describe things that occur.

    There are many mathematical concepts which have no direct application in the observable world. Mathematics has lines that can be separated into infinitely small parts, planes that have zero thickness, lines that have no endpoint and continue out infinitely. These things are straightforwardly handled in the realm of mathematical thought, but none of these things exist in the world we can observe. They can be useful abstractions, but they do not inform reality. You cannot “math” things into existence, be they gods or anything else.

    Math is not “unreasonably effective” at describing the universe at all, it is just a powerful mental tool to give us a springboard for complete science that includes experimentation and verification. In the end all this “sophisticated” theology is just god of the gaps flavored arguments from ignorance.

  36. Posted February 23, 2012 at 7:18 am | Permalink

    Emma Goldman noted that the universe behaves according to laws not because there is a law giver, but because we have come to observe things to be that way.

  37. Matti Sironen
    Posted February 23, 2012 at 7:20 am | Permalink

    Max Tegmark has a rather apropos, non-theistic take on the “unreasonable effectivity of mathematics in physics”, the Mathematical Universe Hypothesis.

    Website: http://space.mit.edu/home/tegmark/crazy.html

    Short exposition:

    http://arxiv.org/abs/0709.4024

    “Full” version:

    http://arxiv.org/abs/0704.0646

    Those papers are a few years old now and I’m not sure if Tegmark espouses them anymore (or to what degree of seriousness he ever did) but it’s certainly interesting food for thought.

  38. Posted February 23, 2012 at 7:25 am | Permalink

    Even if this argument were true, why is it the christian god that is proven by it?

    • GBJames
      Posted February 23, 2012 at 7:28 am | Permalink

      Come on, Graham! What part of “Ergo Jesus” don’t you understand? ;)

    • Mandrellian
      Posted February 23, 2012 at 8:13 pm | Permalink

      It is a hoary old chestnut, that question, but worth continuing to ask because no Sophistimicatered Theologarithmist has ever, once, actually, fracking, answered the bloody thing.

      At best all they can possibly get to (after one grants them significant indulgence) is deism along the lines of “Well, possibly a big, like, Space Dude did it” but they always want everyone to make the same unwarranted leap they’ve made, to “Space Dude = Jesus, human-shaped avatar of Yahweh the Bearded Desert King, Scourge of Serpents, He of the Multiple Genocide, He of the Sociopathic Interest in Your Sex Life”.

      They need to sophisticate a bit harder.

  39. Posted February 23, 2012 at 7:26 am | Permalink

    Contrary to Polkinghorne, the through-and-through scientific comprehensibility of the universe is incompatible with its having been created by a supernatural being.

    If the universe arose supernaturally, rather than by means that natural science could in principle explain, then we have no hope of understanding the universe ever more deeply by investigating it scientifically: we’ll eventually hit a barrier beyond which there’s literally just magic, something that by definition defies naturalistic explanation (and maybe any explanation). Only if the universe is fundamentally non-supernatural — unintended, uncreated — can we hope to delve ever deeper into it.

    So the fact that we’ve achieved ever-deeper understanding of the universe by investigating it is inductive evidence that it isn’t at bottom magical and wasn’t created by a god.

    • Havok
      Posted February 23, 2012 at 8:18 pm | Permalink

      Also, shouldn’t the theist provide reasons why their God would have made that barrier lower than our current investigations rather than higher up?
      Perhaps they’d claim that God continually moves the “magic” barrier away from us, creating more and more detailed scientific explanations for us to “discover”, but then they would surely need to explain that behaviour, and also demonstrate how/why that is different to a world without their God.

      • Posted February 24, 2012 at 6:23 am | Permalink

        I can’t make sense of the idea of a moving magic barrier. It seems to imply the contradiction “The universe was magically created in this particular way and…wait a minute…no, it wasn’t.” One possibility I didn’t mention is that our scientific understanding approaches the magic barrier asymptotically: we get ever-deeper understanding as we continue to investigate, but our marginal gains in understanding keep diminishing. On supernaturalism, that’s the very best we can hope for. On naturalism, we can hope for better — again, very contrary to Polkinghorne’s line.

  40. Posted February 23, 2012 at 7:28 am | Permalink

    ‘Science has disclosed to us a world which, in its rational transparency and beauty, is shot through with signs of mind, and religious belief suggests that it is indeed the Mind of the Creator that lies behind the wonderful order of the universe.’

    I have also noticed that there are a lot of quick things in the universe.

    I conclude that the creator must be really fast.

    And yellow, because there are a lot of yellow things in the universe.

    So we are looking for a quick, yellow god.

    Polkinghorne’s whole ‘argument’ (I am stuck for a more accurate word that would not fall foul of obscenity laws), is based on the absurd non-sequitor that if the universe is ‘regular’ this means it is the creation of a logical mind.

    Just as only a yellow being could create a universe with yellow in it.

    So far Polkinghorne hasn’t even got an argument.

    So there is nothing to refute.

  41. Posted February 23, 2012 at 7:33 am | Permalink

    Although he stops short of “Ergo Jesus”, physicist, cosmologist, and astrobiologist (!) Paul Davies has made rather similar arguments to Polkinghorne’s …

    Where, then, is the evidence of “cosmic purpose?” Well, it is right under our noses in the very existence of science itself as a successful explanatory paradigm. Doing science means figuring out what is going on in the world–what the universe is “up to”, what it is “about.” If it isn’t “about” anything, there would be no good reason to embark on the scientific quest in the first place, because we would have no justification for believing that we would thereby uncover additional coherent and meaningful facts about the world. Experience shows that as we dig deeper and deeper using scientific methods, we continue to find rational and meaningful order. The universe makes sense. We can comprehend it.

    Science is a voyage of discovery, and as with all such voyages, you have to believe there is something meaningful out there to discover before you embark on it. And with every new scientific discovery made, that belief is confirmed. If the universe is pointless and reasonless, reality is ultimately absurd. We should then be obliged to conclude that the physical world of experience is a fiendishly clever piece of trickery: absurdity masquerading as rational order. Weinberg’s aphorism can thus be inverted. If the universe is truly pointless, then it is also incomprehensible, and the rational basis of science collapses.

    Davies is well known for his accommodationism and woolly “spirituality”.

    /@

  42. Posted February 23, 2012 at 7:35 am | Permalink

    Polkinghorne writes ‘ One of the most striking features of the physical world is its rational transparency to us. We have come to take it for granted that we can understand the universe, but it is surely a highly significant factor about it that this is the case. ‘ Polkinghorne specialised in quantum mechanics. He writes about quantum mechanics ‘ We really do not know what the answer is…. The moral, I think, is that explanation and understanding are two different things. We can use quantum theory to explain very successfully a great many things about the world in which we live… But we do not understand quantum theory. ‘

    So is quantum mechanics ‘rationally transparent’ to us or do we not understand it? As it is a major part of Polkinghorne’s case that we can understand the universe , we should be able to understand quantum mechanics.

    It should also be pointed out that while Polkinghorne is happy to live with puzzles in his theology about evil, other religions (‘ I just have to confess my perplexity here. ‘) and our inability to understand a universe which is also transparently rational, he extends no such leeway to non-believers.

    He writes ‘ It goes against the grain for a scientist to be so intellectually lazy. The meta-question of the unreasonable effectiveness of mathematics insists on being answered. ‘

    Atheists must fully account for everything (consciousness, morality, truth, the laws of physics) or they will be accused of having no grounds for denying God, while believers are allowed to retreat into mystery when the going gets tough.

    Polkinghorne writes ‘ The first question is what one makes of the deterministic equations from which the theory of chaos begins. I believe that they must be treated as approximations to a more supple reality…. ‘ He also writes ‘ There may be holistic laws of nature presently unknown to us but capable of scientific discovery….. ‘

    If the universe is rationally transparent to us and we are capable of understanding it, why do we not know what the laws of nature are?

    If we understand the universe, why have we got things so wrong?

    How does Polkinghorne know mathematics is unreasonably effective when our deterministic mathematical equations are only approximations to a more supple reality?

    It seems surprising to say that our deterministic mathematical equations which express the beauty of the laws of nature are a sign of God’s mind, and then turn round and say that they are only approximate and the real laws of nature are of an entirely different kind to the ones we use now.

    • Mandrellian
      Posted February 23, 2012 at 9:28 pm | Permalink

      This, with icing and a cherry:

      “Atheists must fully account for everything (consciousness, morality, truth, the laws of physics) or they will be accused of having no grounds for denying God, while believers are allowed to retreat into mystery when the going gets tough.”

      Abso-freaking-lutely, and it hacks me off no end.

      When an atheist or scientist says, in all honesty, “We do not know the answer,” it’s seized upon in some childish “a-HA!” moment by more theists than you can shake a snake at.

      The believer, as you say, is allowed to just cite “Mysterious ways” – usually whenever something is mentioned that utterly contradicts whatever religious assertion is being made, and that’s the end of it.

      It also hacks me off that allegedly deep thinkers and sophisticates think that rational people are going to buy that shit. What an insult to someone’s intelligence.

      • Posted February 24, 2012 at 4:56 am | Permalink

        Yeah, well, they can A-Ha all they want. All they’re doing is ignoring the history of science. If you look at the accretion of knowledge over the years, it’s basically been a matter of science winning every round of the battle against religion.

        In the meantime, religion has never been able to demonstrate a thing.

        Inductive reasoning suggests that science will keep on going.

        Perhaps the theists have figured this out and are pushing hard in the political arena because the writing’s on the wall as far as the facts go. That’s why I think atheists need to move into politics also.

  43. Chuck
    Posted February 23, 2012 at 7:42 am | Permalink

    I think these arguments are targeted at people who want to believe but see that so much of their religious tradition has been contradicted by what science has revealed. I was one of this hypothetical types when I was religious. Some people get so much emotional satisfaction from religious belief that they reach for possibilities to justify their feelings. In so doing they confuse their hopes as probabilities and fail to see how their explanations explain notging and allow all religions warrant. The explanations only give heat to disagreement with no objective mechanism to silver contradiction.

  44. DiscoveredJoys
    Posted February 23, 2012 at 7:53 am | Permalink

    Laws of physics and mathematics are models of which we use to simplify and understand the universe.

    If all people were to die (so long and thanks for all the diseases) the laws of physics and mathematics would be useless, although the universe and the processes within it would continue.

    On the front foot:

    1) Why is prayer so “unreasonably ineffective”?

    2) Why does knowledge of God explain nothing in the laws of physics or mathematics?

    Ergo no Jesus.

    • infiniteimprobabilit
      Posted February 24, 2012 at 12:38 am | Permalink

      That raises an interesting question – do the ‘laws’ of mathematics exist regardless of any consciousness that is aware of them? At its simplest, does 1+1=2 if there is nobody to count? Did the mandelbrot set ‘exist’ before Benoit Mandelbrot plotted it on a computer? Are the laws of mathematics an invention of intelligence or an inherent property of – what? Space-time?

      I don’t know the answer. (Am I asking a question that philosophers have answered long ago? Probably. I doubt it’s got anything to do with Ergo Jesus anyway).

      • Another Matt
        Posted February 24, 2012 at 6:46 am | Permalink

        I’m not sure they’ve answered it long ago, but I think the question is implied in Pythagoras and explicit in Plato’s Forms.

        I think the best way to look at this is not to talk about “existence” at all, but to follow the logical consequences of definitions, which are inventions (all invention is part discovery, but there’s no reason to attach Aristotelian telos to that idea).

        We have the concept “pi” because first we made the definition “the set of points equidistant from a single point in two dimensions” – pi follows logically, at least in this universe. And I’m not even sure about the idea that pi could have been exactly 3 in another universe, because our pi is even just an abstraction based on flat 2D planes which are themselves abstracted from whatever our space topology actually happens to be (notwithstanding pi’s “spooky” presence elsewhere in math).

      • Posted February 24, 2012 at 4:55 pm | Permalink

        Philosophy of math ticks off a lot of working mathematicians, but in my view fictionalism is actually the way to go in explaining mathematics …

        So no numbers prior to humans (or perhaps other creatures independently) inventing them.

  45. Posted February 23, 2012 at 8:05 am | Permalink

    Yet science itself will not provide its answer, for it is simply content to exploit the opportunities that these wonderful gifts afford us, without being in a position to explain their origin.  Theology, however, can step into the breach.

    Yes it does. And then it makes up answers.

  46. Posted February 23, 2012 at 8:29 am | Permalink

    Jerry, thanks for the plugs. Here s my essay on “Does the Universe Need God?” for the Blackwell Companion on Science and Christianity:

    http://preposterousuniverse.com/writings/dtung/

    And here are some blog posts on the general questions of “why does the universe exist?” and “why are the laws of physics the way they are?”

    http://blogs.discovermagazine.com/cosmicvariance/2007/08/30/why-is-there-something-rather-than-nothing/

    http://blogs.discovermagazine.com/cosmicvariance/2007/11/25/turtles-much-of-the-way-down/

    http://blogs.discovermagazine.com/cosmicvariance/2011/08/11/what-can-we-know-about-the-world-without-looking-at-it/

    • JBlilie
      Posted February 23, 2012 at 12:59 pm | Permalink

      Mr. Carroll, thanks very much for posting those links! I’m reading ‘em now …

  47. moochava
    Posted February 23, 2012 at 8:38 am | Permalink

    The “unreasonable effectiveness” argument is a non-starter. I bet no psychologist has ever wondered why mathematics is so unreasonably effective in their field.

    Worse, math is over-effective, something obvious to anyone whose high school math lessons moved past Euclid into Riemann and Lobachevsky–at most one of them can describe how space works in our universe. Math gives us infinite consistent possibilities, of which only one can be correct for our world.

    So math isn’t unreasonably effective. (How can we even identify its effectiveness as “unreasonable?” Did the psychologists mathematized “surprise” when I wasn’t looking?) And the moment it’s not effective, that, too, is taken as proof of the divine. Particle physics can be reduced to a handful of equations–God’s will. Human thoughts (so far) cannot be–a miracle!

    Also, physics aren’t beautiful. No universe with a neutrino can be beautiful. Everyone knows that!

    • Posted February 24, 2012 at 4:57 pm | Permalink

      It is worse than that, since one can have inconsistent (but non-trivial) mathematics too – there are paraconsistent set theories, for example, and people are working on paraconsistent analysis and arithmetic.

  48. Kevin
    Posted February 23, 2012 at 8:42 am | Permalink

    Rewind absolutely everything you and Polkinghorne have to say on the subject.

    Start from the beginning: “…one needs empirical observations as well.”

    Polkinghorne makes this statement, and then offers not empirical evidence, but arguments. It’s the most-common apologist trap. William Lame Craig has made a career out of this. He’ll make observation after observation about the universe, and then argue that this means god was involved. Nonsense. He hasn’t rejected the null hypothesis – he’s only made a conjecture.

    Evidence is based on objectively verifiable observations that can be reproduced by a disinterested or antagonistic third party. And in this regard, evidence must be used not only to support your contention, but to rule out the null hypothesis.

    For Polkinghorne’s observations to count as “evidence”, they would also have to discount the natural in favor of the supernatural. You can’t just say that this is the solution you prefer, the evidence has to specifically reject the natural.

    There is nothing in Polkinghorne’s arguments that do this. And, as the Catholic Church acknowledges, arguments aren’t evidence because they can be argued. Duh.

    So, every time Polkinghorne makes an argumentative claim, unless he has observational support that overturns the null hypothesis (ie, no god), he’s failed to support his claim.

    • Posted February 23, 2012 at 11:05 am | Permalink

      ExACTly, Kevin! Theologians like Polkinghorne (and under the guise of philosopher, “apologists” like Craig) posit ARGUMENT and then PRETEND they have successfully proffered “proof” that enjoys the crucial support of empirical evidence when in fact they have not. They wax mendacious in the evangelical service of (alleged) God; whether they do so intentionally or under self-delusion I cannot say for sure, but they sure should know enough to know better.

      Unanswered questions “prove” only one thing (a thing nobody disputes): not all questions are presently answered.

      If theologians cannot recognize and understand the vital difference between an ARGUMENT (especially an argument from ignorance) and crucially supportive EVIDENCE, what the heck good are they? (Hmmm…now that I think about it it, what the heck good are they even if they can???)

      • Mandrellian
        Posted February 23, 2012 at 9:47 pm | Permalink

        If theologians could recognise (or, more importantly, if they actually cared about) the difference between rhetoric and evidence, they wouldn’t be theologians.

        Well, they couldn’t be theologians with a straight face anyway.

  49. FBertone
    Posted February 23, 2012 at 8:43 am | Permalink

    Long-time reader, first-time poster here.

    Just to expand some on what’s already been said:

    We’ve spent between several hundred and a couple thousand years (depending upon one’s definition) developing a descriptive tool and now we act surprised when it does what we created it to do.

    When people start talking about the “unreasonable effectiveness” of math I suspect they are headed towards a form of Platonic Realism. Their explanation for why math is so effective will be: because the math is in the universe, ergo jesus. But this is, as another here suggested, mistaking the model for the thing. This is the same mistake that John Searle described in discussing the nature of syntax– the syntax isn’t in the system, it only exists relative to the interests of an observer.

    • Mandrellian
      Posted February 23, 2012 at 9:49 pm | Permalink

      Yep.

      I’d really like it if people like this were able to recognise that physical laws, constants, theories like that of evolution, are _descriptive_ and not _prescriptive_.

  50. yairr
    Posted February 23, 2012 at 8:44 am | Permalink

    The great physicist Murray Gell-Mann made an interesting TED Talk lecture about why, he suspects, mathematical elegance is important in fundamental physics.

    I don’t believe I can do his explanation justice, so I just recommend hearing what he has to say.

    Yair

    • Posted February 23, 2012 at 11:12 am | Permalink

      Delightful video, THANKS for the tip on that, Yairr!

      I also highly recommend Murray Gell-Mann’s book, THE QUARK AND THE JAGUAR.

  51. Posted February 23, 2012 at 9:13 am | Permalink

    Mathematician John Allen Paulos addresses this issue in his book Irreligion: A methemtician explains why the arguments for god just don’t add up (p. 127-132).

  52. Captain Howdy
    Posted February 23, 2012 at 9:19 am | Permalink

    A number of comments have pointed out that Polkinghorne’s is an elaborate God of the Gaps argument. (Actually it’s not that elaborate.)

    Note the number of times he asks “Why?” Polkinghorne doesn’t merely pose the inevitably unanswerable question, he insists that it is “intolerably intellectually lazy not to seek to pursue this question.”

    Of course one can ask “why” with infinite regress, and therefore one will eventually, inevitably run into a gap in our understanding. You can either continue the exploration, thereby increasing our body of knowledge about the universe, or throw your hands up and declare God dunnit.

    Who’s the intellectually lazy one here?

  53. Nicolas Perrault
    Posted February 23, 2012 at 9:32 am | Permalink

    “The unreasonable effectiveness of mathematics.”

    This argument fails to impress me. Math works for two main reasons:

    The first has to do with rigour. The axioms are carefully laid and are consistent as far as we can see. It is thus possible to spot faulty thinking even in a long chain of arguments and algebraic manipulations. If applied correctly, mathematics will not lead to contradictions or paradoxes. (In the unlikely event that this happens mathematicians won’t hesitate to change the axioms and definitions to avoid all contradictions and paradoxes. For example this was done when mathematicians were confronted with paradoxes in set theory such as Russell’s paradox).

    The second: we are free to adjust the axioms and definitions so that they very closely match whatever we observe. From the observation that candies do not spontaneously appear, disappear or merge we can account for them by using the simple rules of arithmetic. If the axioms of Euclidian geometry fail near a massive body, we’ll use non-Euclidian geometry instead.

    The simple fact that we exist is to me conclusive evidence that there is some kind of order out there. Math is now so advanced and powerful that there is usually a mathematical way to at least partially encapsulate what we observe.

    In conclusion, the unreasonable effectiveness of math is to me much less puzzling than would be its chronic ineffectiveness. To see a proof of God in the effectiveness of math is a “sophisticated” form of wishful thinking.

  54. Ian Atkinson
    Posted February 23, 2012 at 9:33 am | Permalink

    When it is known what God is made of, how he functions as an intelligent being, how he can be detected by senses or instrumentation why he is eternal and the Universe is not, God will be ‘science.’ Until then he will remain a leftover from an ancient Canaan pantheon.

    When any trace of Jesus can be found, or any record of his existence contemporary to his life, or any record before Paul of Tarsus had his vision, then Jesus will also be ‘science.’

    For millennia philosophers have been using tricks of language; metaphors, use/mention errors, swapping existence of ‘concept of object’ for existence of the ‘object,’are all mealy mouthed tricks to muddy the waters between fact and non-fact. Thank goodness we’re gradually entering an age of objectivity.

    A musky smell and rubbish scattered around a garden may or may not be proof of a fox, but it isn’t proof of anything to somebody who’s never seen or heard of a fox. Empirical Proof of God’s actual existence is needed *before* aspects of the Universe can be taken as proof of him having been there.

  55. Daniel Engblom
    Posted February 23, 2012 at 9:34 am | Permalink

    Richard Carrier has posted a collection of his refutations on these issues to these “arguments”. Anyone want to check them out? I’m not qualified, but I like Carrier already (has had many funny, clear and instructive talks over at skeptikon – here’s the youtube channel: http://www.youtube.com/user/HamboneProductions#g/u)

    Here’s the blog post:

    http://freethoughtblogs.com/carrier/archives/373/

    Opinions? Think this is enough to refute Polkinghorne’s nonsense?

    • Kevin
      Posted February 23, 2012 at 10:17 am | Permalink

      He doesn’t say it outright, but it looks like a Bayes analysis.

      But again, in my opinion he misses the mark because evidence cannot be used to justify position “A” and position “not A” at the same time.

      All of Polkinghorne’s “evidence” — whether or not we agree with his suppositions — can be used to support the following “not A” contention:

      “An all-natural universe arose from all-natural processes that continue today, and can be best understood via model-dependent realism.”

      Polkinghorne and all other apologists have to show why their “evidence” (which isn’t evidence but a series of observations) reject this “not A” hypothesis. They can’t.

      The very best they can do with this line of reasoning is a tie. Neither “A” nor “not A” are supported. And in this setting, the tie goes to the null hypothesis (ie, no god). Not to some vague deism that leads to “ergo, Jesus”. The tie goes to “all-natural processes.”

  56. Xuuths
    Posted February 23, 2012 at 9:53 am | Permalink

    Polkinghorne refuses to acknowledge and state that math has evolved.

    Imagine two early people discussing counting:

    Person A: Counting is simple. 1 apple plus 1 apple equals 2 apples.

    Person B: OK, let me try. 1 dirt pile plus 1 dirt pile equals 1 bigger dirt pile. This is complicated.

    Person A: No, you’re just stupid. Counting is simple.

    catholic Person C: 1 father plus 1 son plus 1 holy ghost equals 1 god.

    Person A: Get out of here you nut!

    Math evolved.
    What didn’t work was discarded. What turned out to be inaccurate was discarded. What was found to be needlessly complicated (a simpler solution was discovered) was discarded. What provided less accurate results was discarded.

    Of course over time math got better, more accurate, and with more elegant solutions. That would happen in any system — even intermittently random ones would eventually yield some kind of math formulas.

    Sheesh, Polkinghorne. Get a grip!

  57. Neil
    Posted February 23, 2012 at 10:00 am | Permalink

    Believers want it both ways. If the universe is comprehensible, it must be god. If the universe is incomprehensible, as in how did that electron go through both slits, it must be god.

    • Circe
      Posted February 23, 2012 at 10:14 am | Permalink

      But surely the only(TM> explanation for the electron’s behavior can be that god personally chaperoned it through both the slits? Ergo {Krishna, Shiva, Allah, Quetzalcoatl, Zeus, Thor, Ganesha, Great Juju at the Bottom of the Sea, Jesus, Jehova}!

      • Yiam Cross
        Posted February 23, 2012 at 5:14 pm | Permalink

        You miss the point. While Allah and Quetzalcoatl are chaperoning the electron through the slits, Gnaesha, Thor, Zeus and all the rest are busy keeping transatlantic flights in the air, hard disc drives functioning correctly, church spires from toppling over etc etc. No wonder we need so many gods and they really do have to be supernaturally powerful. It’s not as easy a life as you might imagine. Ask Bruce (Almighty). Or maybe Morgan Freeman.

      • sasqwatch
        Posted February 23, 2012 at 5:20 pm | Permalink

        No. Buddha.

        Because his mother had two slits, including the one he was born out of. Some guys can’t theolosophize for shit.

        • Circe
          Posted February 23, 2012 at 11:30 pm | Permalink

          Let’s leave poor Siddhartha Gautama out of this. The man was apparently fed of the ritualistic religions of the day and tried to start an agnostic religion to combat it, and even apparently had the honesty to point to his followers that he was no god, offering as proof his impending death through the usual cause of old age and disease, and what do his followers do after his death? Why, break up into two sects, with the majority sect deciding to worship the guy as a god, complete with rituals. The guy must have facepalmed if there was an afterlife.

          • sasqwatch
            Posted February 24, 2012 at 3:37 am | Permalink

            :-) A two-slit experiment gone horribly wrong.

  58. Circe
    Posted February 23, 2012 at 10:07 am | Permalink

    The universe can be comprehended by the rational faculties of humans, and its workings appear to adhere to laws of physics.

    Much of our understanding of the universe is expressed through mathematics, which is “unreasonably effective” in encapsulating what we discover about nature.

    Ergo Jesus.

    This always bothers me. Why not “Ergo {Krishna, Shiva, Allah, Quetzalcoatl, Zeus, Thor, Ganesha, Great Juju at the Bottom of the Sea}?

  59. Posted February 23, 2012 at 10:12 am | Permalink

    Humans have fallen in love with an indifferent universe after existing in it a bare fraction of a second. Except for one tiny planet, the universe appears to be an inhospitable, deadly place and certainly not evidence of a loving deity(s). Life as we know it is an accident. If a god is responsible, he is selfish, cruel, stingy, fickle, arrogant, dishonest, and an unintelligent designer.

    • Circe
      Posted February 23, 2012 at 10:16 am | Permalink

      It does not seem to be such a deadly place for stars, which suggests that if there is a god, she must be a star.

      • Kevin
        Posted February 23, 2012 at 10:20 am | Permalink

        Yes, but since fundamental process of stars is to convert H to He (and other elements), god must be inordinately angry at H.

        • Circe
          Posted February 23, 2012 at 10:33 am | Permalink

          ‘He’ is a false god, so she needn’t be anything more than bemused by “H(e)is” antics.

    • derekw
      Posted February 23, 2012 at 3:19 pm | Permalink

      So you say…while typing on your 17″ laptop sitting in a warm room on your Lay-Z-Boy sipping a hot latte while watching NCIS on your 60″ LED LCD.

  60. Posted February 23, 2012 at 10:23 am | Permalink

    I’ll say it again: the utility of mathematics makes the possibility of some “designing force” to be a conjecture with considering…nothing more than that. It has nothing to do with any deity that humans have conjured up.

    Why not some other deities conjured up by beings in another galaxy or something no being anywhere has conjured up?

    Those who claim that this is any evidence for any human god hasn’t really accepted the Copernican principle that there is nothing “special” about the earth, humans, etc.

    I think that Sean Carroll’s point of view makes far more sense.

  61. PoxyHowzes
    Posted February 23, 2012 at 10:30 am | Permalink

    I’m a true believer that “mathematics” doesn’t exist in any “real” sense.

    a) None of us will ever see a line exactly Pi units long, or 10/3 units, or SQRT(2) units. That illustrates that maths is as much a notational tool as a descriptive one. Check out Newton’s versus Leibniz’s calulus.

    b) Almost all useful “mathematical” applications these days are successive approximations built by numerical methods from engineered calculators.

    Successive approximations was the method Aristotle used to approximate Pi. Successive approximations is the foundation of “The” calculus, whether Leibniz’s or Newton’s. Millions or billions or trillions or more of successively calculated approximations, and not a whit of Newton’s formulas, got us to the moon.

    So no, Mathematics is not “unreasonably” effective at describing the real world. Mathematics is (by definition and by practice) an idealized representation of scattergrams of observations of the real world.

    • Circe
      Posted February 23, 2012 at 10:38 am | Permalink

      “None of us will ever see a line exactly Pi units long, or 10/3 units, or SQRT(2) units. That illustrates that maths is as much a notational tool as a descriptive one. Check out Newton’s versus Leibniz’s calulus.”

      Why not? just draw a line, and define its length to be units. Of course, you may never be able to draw an ideal line (papers is not really a planar, if you look close enough), but that’s beside the point.

      • Circe
        Posted February 23, 2012 at 10:40 am | Permalink

        “Successive approximations was the method Aristotle used to approximate Pi.”

        As far as I know of the history of trigonometry, the earliest estimates of the value of pi were made my Archimedes, and then improved further by Aryabhata (whose value remained the best computed for about a millennium after him), and not be Aristotle.

  62. Circe
    Posted February 23, 2012 at 10:40 am | Permalink

    “Successive approximations was the method Aristotle used to approximate Pi.”

    As far as I know of the history of trigonometry, the earliest estimates of the value of pi were made my Archimedes, and then improved further by Aryabhata (whose value remained the best computed for about a millennium after him), and not be Aristotle.

  63. abb3w
    Posted February 23, 2012 at 10:58 am | Permalink

    The reason mathematics is effective at describing the universe is because mathematics is effective at describing ANYTHING, per Gödel. (The reason mathematics seems efficient at describing parts of the universe is because the more efficiently solvable math is easier to recognize.)

    The reason human reason can handle mathematics is a result of the potential of mathematical systems to self-model, per Turing.

    (Alas, I didn’t get the chance to discuss this line of reasoning with John Lennox when he wandered through town; there weren’t enough slots in the small group discussion he had with some of the local student atheist and Christian groups.)

    I’d also respond to Jerry’s implicit question:

    We don’t understand why there are physical laws that behave with regularity

    …by suggesting again that we don’t even know that there ARE such laws. Rather, this is taken as a primary philosophical assumption, where the alternative of Refutation is fully as consistent with the evidence. In the case of the latter, any appearance of order/pattern/law is merely a local illusion, inevitable under Ramsey’s Theorem in a sufficiently large sea of chaos. This leads to no implications considered “interesting”, so the alternative of Assertion is taken… in which case, the rules themselves (or the minimal meta-rules generating the other rules) are the ultimate “uncaused cause”, and the assumption gives the mathematical potential inference as to which are more likely. And the Second Law of Thermodynamics appears uninterested in human worship.

    Or, from another angle: it’s just an assumption, without priors. Instead of saying God implies Rules, and thus in effect needing God and Rules for the philosophy, we simply assume Rules directly.

  64. Posted February 23, 2012 at 11:29 am | Permalink

    Maybe I’m missing something, but what, precisely, IS the argument? Neither statement is an argument, they are only observations. We need a testable theory, or a syllogism, to plug them in to. It’s fundamentally irrelevant whether either statement is true unless there is some persuasive context.

    Wait, let me guess…we’re just supposed to assume that if something is mysterious, therefore it is divine?

  65. Dan L.
    Posted February 23, 2012 at 11:49 am | Permalink

    The universe can be comprehended by the rational faculties of humans, and its workings appear to adhere to laws of physics.

    As others have already pointed out, what we call “laws of physics” adhere to the workings of the universe, not vice versa. There’s also a little question begging going on in the “comprehended by the rational faculties” bit. Suppose the universe such that our “rational faculties” could not comprehend it. By what lights could we insist that our faculties are really rational and it is the universe (not our faculties) being wayward?

    That is, “rational faculties” can only be defined relative to a comprehendable universe in the first place. This is ignoring the fact that there seem to be rather sharp limits to our comprehension: no one finds quantum mechanics intuitive and only a very few human beings understand general relativity and a lot of other concepts we use to understand the universe. Our best models predict only 1% or so of the energy content of the universe and also make a few other predictions that don’t seem to hold. For the most part, every time we claim to comprehend any part of the universe we’re actually pretty wrong about it.

    Much of our understanding of the universe is expressed through mathematics, which is “unreasonably effective” in encapsulating what we discover about nature.

    The only comprehensive definition of mathematics I’ve ever heard is “the study of patterns.” If you look across time and cultures at different traditions of mathematics you get different notations, different terminologies, different approaches, but all bearing on the same patterns. That is to say, as long as there is such a thing as patterns there is such a thing as mathematics. This is exactly the “laws of physics” error from above but applied to abstractions rather than the universe. It’s not that we have this thing called mathematics and for no explicable reason it seems to be really good for describing patterns. It’s that we saw patterns and developed notations and other systems to describe those patterns.

    To say that this is somehow “unreasonable” is to suggest that a universe without patterns is a more “reasonable” possibility. The onus is on Polkinghorne to explain what this even means. Is a patternless universe somehow more intuitive or obvious than a patterned one? Is it even possible? Intelligence itself seems to manifest as a capacity for recognizing patterns, what would such a faculty amount to in a patternless world?

    Other people already mentioned Douglas Adams’ intelligent puddle. They were exactly right, Polkinghorne is make the same arguments as the puddle.

  66. Bjarte Foshaug
    Posted February 23, 2012 at 11:58 am | Permalink

    What god in his right mind would make the two most important constants of mathematics (pi & e) two endless strings of random numbers, thus making calculations so much more difficult than they could have been, if they were free to vary at all? In fact, why not make it so that every natural number could be written as a fraction of two integers (as the Pythagoreans orginally believed). It would certainly make my day easier.

    • infiniteimprobabilit
      Posted February 24, 2012 at 12:52 am | Permalink

      I kinda tend to think that would be physically impossible, certainly in the kind of 3-dimensional space we inhabit.

      I’m no mathematician, but I think that if space was curved in such a way that PI=3 for example, then *we* would be physically impossible. Even for God. :)

      • Bjarte Foshaug
        Posted February 24, 2012 at 5:36 am | Permalink

        Which is why I added the qualifier “if they were free to vary at all”. If they are not, there is nothing for God to explain. ;)

  67. Joshua
    Posted February 23, 2012 at 12:05 pm | Permalink

    I realize this has already been commented on, but I’d like to add my two cents on Jerry’s comment:

    “We don’t understand why there are physical laws that behave with regularity”

    This is, of course, an assumption as mentioned earlier.

    Furthermore, the usage of the term “physical law” is not always clear. Most of us are familiar with Newton’s laws or the Laws of Thermodynamics. Outside of these rather reserved uses of the term “law”, we don’t see it that often in Physics. Examples: “Maxwell’s Equations”, “Theory of Relativity”, “Quantum Theory”, and so on. The usage seems to have fallen out of fashion since around the time of modern physics (1900 or so).

    What’s more, the term “law” is actually misused (and quite knowingly) by physicists. The examples I can think of are a bit esoteric, but none-the-less here you have them: “Ohm’s Law” in electricity, “Hooke’s Law” in elasticity theory; these are in fact not “laws” in the common sense, but we don’t change the name (out of tradition I suppose?).

    An aside: in classical mechanics there is also a term “canonical variable” which appears in regular usage. My guess is that the terms “law” and “canonical” are merely vestiges of the religious past of which physics was born.

    • Posted February 24, 2012 at 5:01 pm | Permalink

      They are vestiges, but there’s an important distinction to nevertheless make.

      All of what you gestured at are actually law *statements*. What I think really matters for the discussion are the real patterns the law statements *refer* to.

      (A theory is just a system of said statements, closed under a deduction relation.)

  68. Kevin O'Neill
    Posted February 23, 2012 at 12:39 pm | Permalink

    Polkinhorne’s argument is so evidently feeble that it does not deserve serious debate: his resembles the argument that the conditions for evolution of life are so unlikely that they could not have occurred by chance. Just applying the same flawed logic to a mathematical argument to claim that we are ‘lucky’ that the maths fits the model when of course it does: the maths comes from the model. A tired argument in a new(ish) dress!

  69. JBlilie
    Posted February 23, 2012 at 12:51 pm | Permalink

    Polkinghorne is all wet! Those points he made simply prove that Ceiling Cat exists, for Ceiling Cat’s sake! Polkinghorne: Show me where my logic is wrong on this! ;^)

    • Yiam Cross
      Posted February 23, 2012 at 4:48 pm | Permalink

      Er, we have a photo of ceiling cat. Where’s the photograph of god, jesus or even one of the many rooms in his father’s mansion.

  70. Another Matt
    Posted February 23, 2012 at 1:20 pm | Permalink

    There are also these lines of thought to consider:

    http://en.wikipedia.org/wiki/Is_logic_empirical%3F

  71. Posted February 23, 2012 at 2:06 pm | Permalink

    I’d say it’s ascertainment bias. He spots the neat solutions and marvels at them but seems to ignore other observations, such as the difficulty constructing a Grand Unifying Theory. Surely if the Universe was such a clean and mathematical construction of a higher power for us to discover, such a thing would just pop out? It’s the same with all the arguments from design – they focus on the designed-looking end of the spectrum and ignore the rest.

  72. MadScientist
    Posted February 23, 2012 at 2:19 pm | Permalink

    I wonder what Polkinghorne’s criteria are for unreasonable effectiveness. He sounds like a babbling moron to me.

  73. Yiam Cross
    Posted February 23, 2012 at 4:46 pm | Permalink

    Maths is unreasonably effective in describing the universe as we know it in the same way a roof is unreasonably effective at keeping the rain out. Because that’s what we built it to do.

    And if maths came from god, how come I have to fix my own roof? Jesus, bring your crawling ladders!

  74. Peter
    Posted February 23, 2012 at 6:25 pm | Permalink

    Wigner’s article is easily obtained on the internet. It is hard to detect whether even a single correspondent so far has read it. That we are able to discover physical theories with extraordinary predictive powers is even more mysterious now than when he wrote the article, with something like at least 10 digit accuracy in some cases. Wigner marvels at this (weaker then).

    But his title refers to something different, the fact that mathematical theories discovered in a purely aesthetic pursuit turn out to be a large component of these theories, particularly the application of complex separable Hilbert space.

    I agree with Shallit that this is not quite as unreasonable as Wigner thinks, but not for Jeff’s reason. It seems to me that mathematics is an extraordinarily large extension of logical methods, but still logical methods. There may be some disagreement among philosophers about the foundations of mathematics, but I don’t think my statement above, in the context of the relation to physical science, is affected much by that. And I don’t think anyone would write an article entitled “The Unreasonable Effectiveness of Logic in Physics”.

    So I think the real mystery is what is in the first paragraph above. Writings in recent decades by Putnam and Tegmark are ones of interest here. For example, the latter physicist at MIT maintains that the physical universe is not merely modelled by a mathematical structure, but actually IS such a structure.

  75. Diane G.
    Posted February 23, 2012 at 6:52 pm | Permalink

    (subscribing. Wow, when JAC asks his readers to come through for him, do they ever!)

  76. Terry
    Posted February 23, 2012 at 8:21 pm | Permalink

    It’s simpler and more elegant to see that we are children of the Big Bang; the result of uncountable molecules bouncing off each other for about 14 billion years – no soul nor gods necessary or required

    • infiniteimprobabilit
      Posted February 24, 2012 at 12:57 am | Permalink

      You realise that every one of those bounces had to be precisely calculated to the nth degree to arrive at you? That if even one of those bounces had been ‘off’ you wouldn’t be here? How astronomically improbable is that? Ergo Jesus! ;)

    • Posted February 24, 2012 at 5:05 am | Permalink

      Funny coincidence. There’s a minor skirmish on LinkedIn based on the notion of the ‘hand of god’ making the galaxies accelerate away from one another: http://lnkd.in/3JBWZQ

      Warning: the logic is so twisted it may make your head explode.

  77. Dave Ricks
    Posted February 23, 2012 at 8:51 pm | Permalink

    How would anyone judge math to be “unreasonably” effective? Judged compared to what, and judged by what criteria? I see that as an argument from incredulity, a feeling of “gee whiz” and that’s that.

    But more importantly, I can set aside what I know about math and physics to focus on history. In the video posted here a few days ago, where Neil deGrasse Tyson talked about “The Perimeter of Ignorance”, when Newton could not model the stability of orbits, Newton deferred to a god. But today, when Polkinghorne sees what we can model, he claims that as evidence for a god. This is some strange history.

    It’s simply wrong to conflate these opposite ideas about gods in sentences that use one name capitalized “God” as a proper name as if there’s only one and we know which one of these opposites we’re talking about.

  78. Bryan
    Posted February 23, 2012 at 11:11 pm | Permalink

    Polkinghorne: “The universe can be comprehended by the rational faculties of humans, and its workings appear to adhere to laws of physics.”

    Harris: “I have often wondered why walking works. Why is the world organized in such a way that we can walk upon it?”

    Harris’ essay is pure gold – it’s my favorite thing that Jerry has ever linked to:

    http://www.edge.org/3rd_culture/coyne09/coyne09_index.html

  79. dunstar
    Posted February 23, 2012 at 11:27 pm | Permalink

    As Feynman said, at the very bottom, at the most fundamental particles of nature there are no more gears and wheels inside them to explain how they behave. It can no longer be described by a mechanistic process. It’s just the way it is! That’s where “why” questions end! At that level, “why” questions are non-sensical. lol.

    • Another Matt
      Posted February 24, 2012 at 6:31 am | Permalink

      Also, this is where any hope for Aristotelian physics has to be abandoned completely – there’s no meaningful form/content distinction at that range, and at that level it’s only possible to know particles by how they behave rather than what they’re “made of.” Or to put it even more strongly: what they are is how they behave.

  80. Posted February 24, 2012 at 3:43 am | Permalink

    It strikes me that you can’t claim that the “unreasonable effectiveness” of mathematics is evidence for God and then claim that our inability to find mathematical models for things such as love is also evidence for God.

  81. Torbjörn Larsson, OM
    Posted February 24, 2012 at 6:17 am | Permalink

    While the logic of Polkinghorne’s overall argument fails on the fallacy of a false dilemma, you can write essays on those questions.

    In fact, my comments over the years may constitute one. Here I will confine myself to narrow answers, and comment on the broader aspects afterwards:

    The universe can be comprehended by the rational faculties of humans, and its workings appear to adhere to laws of physics.

    These have straightforward answers:

    – Evolution provided those faculties against a structured enough physics.
    – A structured enough physics appears in multiverses, which is a generic physics.

    The latter fails Polkinghorne’s argument.

    Much of our understanding of the universe is expressed through mathematics, which is “unreasonably effective” in encapsulating what we discover about nature.

    This puts up two sequential blocks for Polkinghorne:

    – We can observe that empirical physics applies mathematics by empirical heuristics. (Say, when doing quantization, which can’t be axiomatized.)
    – We can observe that mathematics is quasiempirical. This is seen from the problems of axiomatizing how to do proofs, which are currently heuristics developed by breaking down the steps to mutually agreeable parts. Over how to check them for errors. To such quasiempirical mathematical objects as Chaitin’s constant.

    To overcome these blocks would take Polkinghorne to a one-one map mathematics physics, if not nature but “what we discover about nature”.

    Then he has to come up with a measure for effectiveness, and a mutual agreeable definition of “unreasonable” despite we are lacking a comparison.

    My 2p. (Some or all of these problems have been commented on already, I believe.)

    —————————————-

    Broader aspects:

    – We don’t need multiverses. It has been long known that chaos has to encompass non-bounded sets of order, from Ramse theory. An infinite universe, which it looks to be, with chaos would hence be expected to have infinitely many ordered observable universes in it.

    – We don’t need “laws”. They are expressions of symmetry and spontaneous symmetry breaking, the latter which happens non-conserved systems are replaced with conserved systems (having energy) that then can minimize energy.

    It is a process that we can observe all the time. As Stenger notes in “God – the Failed Hypothesis” even if we have no concept of “nothing” initial chaos has a maximum amount of symmetry. Its laws look different, but it has in principle at least as many _general_ laws. (Without structure formation, there are fewer cases of, say, biological evolution. =D)

    My notion is that it has to have that, since we see more symmetry looking back, and religious insist on that we come from “nothing”.

    Notably, creationists do exactly the same mistake here as they do on information in the genome: a randomized genome has maximum information (Kolmogorov complexity); a randomized universe has maximum order (sets of symmetries).

    – Mathematicians and philosophers of mathematics would probably mostly not agree with what is observed, but hold out for Platonian ideals in the form of unobservable mathematical objects. I think that is a form of dualism.

    Tegmark makes mathematical objects = physical objects on the basic level to have a map mathematics nature. But then his multiverses looks decidedly different from randomized physics, and it is unparsimonous on physics.

    But that is philosophy of theological character which goes against observation so far. And it has nothing to do with the two blocks Polkinghorne has to overcome first.

    • Torbjörn Larsson, OM
      Posted February 24, 2012 at 6:30 am | Permalink

      I note that I use “order” in two senses. Not randomized/structured is the usual sense, “laws” is the physics (or maybe creationist) sense. Oh, well.

    • Torbjörn Larsson, OM
      Posted February 24, 2012 at 6:32 am | Permalink

      Oops. It’s _Ramsey theory_.

    • sasqwatch
      Posted February 24, 2012 at 12:26 pm | Permalink

      Really cool stuff. I’ll put in a plug for Stenger’s “The Comprehensible Cosmos” as well, for a bit more fleshing out of the concept of “symmetry-breaking” and how various symmetries lead to Laws.

      The appendices of that book are for the more adventurous, but are still accessible to the advanced/determined layperson. A very concise manual of standard model physics, IMHO.

  82. Michael
    Posted February 24, 2012 at 7:24 am | Permalink

    Two kinds of responses have come up to Polkinghorne’s argument. Jerry makes both of them, but in doing so he’s trying to have his cake and eat it too.

    First answer, exemplified by Sean Carroll: we can’t answer the question why the universe is comprehensible. The best answer to this is: it just is. (See his article “Does the Universe need God?” which he helpfully linked to above.)

    Second answer: this is a god-of-the-gaps argument and science will eventually explain why the universe is comprehensible, etc.

    Thus, heads Jerry wins and tails Polkinghorne loses: if there is no explanation we give Carroll’s response, but if there is an explanation we’re safe too.

    But you can’t have it both ways. If the correct answer is just “that’s how it is” then the correct answer can’t also be “there is an explanation but we haven’t got it yet.” And it is Carroll’s response that is the more reasonable one. Any answer along the second lines would presuppose that which it tried to explain. If it is really an explanation of the regularity and intelligibility of the universe, it will have to make use of some general principles that govern the universe and make the explanation work. And these principles will not be explained by the explanation without circularity, since the explanation requires them. (This is really just Hume’s argument about inductive reasoning, slightly modified.)

    I cannot see (can someone help me?) how a multiverse explanation of the particular laws of our universe can get off the ground without some more general laws that govern the whole multiverse — certainly not in those versions of the multiverse hypothesis where it is claimed to be a testable theory. (If it is just a mathematical formalism representing a space of all possible universes, then the claim that it is real and not merely possible is as metaphysical a posit as the posit of the theistic God.)

    • Posted February 24, 2012 at 7:50 am | Permalink

      I think the most reasonable answer is “We don’t know at the moment, and we don’t know if we can know in the future.”

      History shows that some things we thought we couldn’t figure out eventually do get figured out. Even in those cases where we still haven’t got answers, we at least understand the question better, which is usually necessary before the breakthrough that gives an answer.

    • Another Matt
      Posted February 24, 2012 at 7:55 am | Permalink

      I cannot see (can someone help me?) how a theistic explanation of the particular laws of our universe can get off the ground without some more general laws that govern the heavens.

      In any case, the place Polkinghorne’s argument gets off track is to assume that consciousness and reason are themselves things that exist somewhere or in some way separate from the universe. He might have a point if brains were not computers that operate according to the same laws of physics. Polkinghorne, if I remember correctly, claims not to be a dualist (when he dies, God will remember his neuronal pattern -> “his soul,” and will recreate it in a new body). But there’s an implicit dualism in his argument, along the lines of Feser’s common comment that evolution couldn’t have given us our powers of reason because reason is another essence separate from the base intelligence other animals have.

      It’s all similar to Plantinga’s argument that if there is no god we have no reason to have confidence in our reasoning abilities, because evolution could have delivered us brains that were great at helping us reproduce but poor at finding out what is true. I don’t think that one is even worth discussing here.

  83. Kharamatha
    Posted February 24, 2012 at 8:23 am | Permalink

    Why can’t I believe it’s not butter?

    Why is this Sparta?

    Why did the chicken cross the road?

    Ergo Jesus.

  84. DV
    Posted February 24, 2012 at 3:45 pm | Permalink

    It has been more than 150 years since Darwin’s “On The Origin Of Species…” and still we have smart people struggling with big questions as if evolution has not been discovered yet. It is amazing how much the explanatory power of evolution is overlooked. Why do we find the world intelligible? God is not the answer. Evolution is.

    • Another Matt
      Posted February 24, 2012 at 3:53 pm | Permalink

      +1

      It’s the same in principle as the question of why there’s an “unreasonable” amount of delicious and nutritious food.

  85. Posted February 24, 2012 at 4:03 pm | Permalink

    I’ll read the rest of the comments and then remark again as necessary, but the single biggest mistake in the premisses (ignoring the silly leap in the conclusion) is assuming mathematics applies to reality. It doesn’t; instead it applies at second hand, to our *ideas* about reality. It is thus no mysterious than using language. Some think the realism of everyday language is evidence for their pet superstition, which is ridiculous too, but at least then we’ve moved away from a crucial error.

    As for lawfulness, this is crucial. It is a key metaphysical finding both supported by and presupposed by scientific research. However, and most damming for the theist (including the deist) is that *lawfulness is incompatible with theism*.

  86. Leigh Jackson
    Posted February 26, 2012 at 2:18 pm | Permalink

    Mathematics applies to the thinking and non-thinking physical world. No mysticism required.

    The world is not shot through with mind because it is shot through with mathematics. Physical order does not demand an ordering mind. Perceiving order does. Evolution can produce exquisite order without need of mind.

  87. Moss McCarthy
    Posted March 23, 2012 at 10:58 pm | Permalink

    The best proof of God in my view is an argument from epistemology. In order to know anything there must be something to know (experience). But whether the experience is real or illusory depends upon an act of knowledge. Experience is undefined until after the act of knowledge. But the act (or power to obtain knowledge) itself absolutely must exist or else there can never be knowledge. The act is necessarily indefinable. Please note well the difference between undefined and indefinable. That act is the light of the world. Therefore if science exists then the reality of Christ is absolute. Better said, the act (God, Christ or the Creator) exists in a way utterly different to all other existents.

    • Posted March 24, 2012 at 5:44 am | Permalink

      Do you actually buy it yourself, though? It has the obvious problem in the claim that the only knowledge is that which can be absolutely deduced. This is simply not the case – any evolved being will gain knowledge by not being killed by its environment (and whether that knowledge is stored in the genes or brain doesn’t matter there), so practical, useful knowledge can in fact be tentative. And everything a human claims to know is in fact tentative. The probability against may be laughably negligible, but 0 and 1 aren’t probabilities.

    • gbjames
      Posted March 24, 2012 at 6:35 am | Permalink

      Come on, Moss, get real. As in “reality”. Having an delusion, is not knowledge in any meaningful sense of the word. Knowledge is measured by the relative correspondence of a mental model of things to the real universe. You’re “best proof” is no better for Jeebus than it is for Huitzilopochtli or the Flying Spaghetti Monster.

    • Steve
      Posted March 24, 2012 at 6:39 am | Permalink

      Well this “best proof of God” failed to prove such to me.Are you saying there can’t be any man-made fictions?

    • bernardhurley
      Posted March 24, 2012 at 10:03 am | Permalink

      Moss McCarthy, this lookslike incoherent garbage to me.

      In order to know anything there must be something to know (experience).

      Without the word “experience” in parentheses, this would just be an uninteresting tautology. However I wonder why you put it there. Experience is not knowledge, unless you are making a Cartesian point about knowing you have the experience. Experience is one of the ways we justify knowledge claims.

      But whether the experience is real or illusory depends upon an act of knowledge.

      The word “know” is a verb as is “eat”. But you are merely being deluded by the grammar of the language if you think there is such a thing as an act of knowledge in analogy with an act of eating. You might more plausibly speak of an act of knowing – e.g. if I am knowing something until I forget it. That would be a rather strained use of language, but it has the advantage of making sense

      All experience is real; the question is when and under what circumstances it justifies knowledge claims. For instance if I had filled myself with drugs or gone into a trance or have certain brain disorders then it may not.

      But the act (or power to obtain knowledge) itself absolutely must exist or else there can never be knowledge.

      You are conflating two things here. First an act of knowledge, which, as I have said, is merely a confused use of language and a power to obtain knowledge. Now I have a power to obtain ice cream, when I can afford it. That does not mean I will obtain ice cream. For instance the shop may have run out or the shop keeper might cheat me. It is possible, but unlikely, that I have always been cheated when I exercise my power to obtain ice cream and have never tasted ice cream in my life. The same could be said for knowledge.

      If by “there never can be knowledge”, you mean that there can never be certain knowledge then that is correct. All we can hope for is justifiable knowledge claims. In view of this, there is a moral imperative to make sure these justifications are as good as possible.

      The act is necessarily indefinable. Please note well the difference between undefined and indefinable.

      The same conflation: The act of knowledge is indeed undefinable unless you mean by it the power to obtain knowledge when it becomes quite easy to define.

      That act is the light of the world.

      Doesn’t it seem strange to you that you can kow this about something you cannot define? Or perhaps it’s just a metaphor.

      Therefore if science exists then the reality of Christ is absolute.

      Now we see the point of the metaphor. The same metaphor has been used for Christ. But you really can’t argue two things are the same because the same metaphor has been applied to them. Incidentally, schizophrenics regularly argue like this.

      Given your conflation of an act of knowledge with the power to know something, your argument seems to boil down to the following (with my comments in square parentheses):

      1. Science exists.
      2. For science to exist absolute knowledge has to exists.[untrue]

      Therefore from 1. and 2.:
      3. Absolute knowledge exists.

      4. For absolute knowledge to exists an undefinable act of knowledge must absolutely exist. [incoherent]

      Therefore from 3. and 4.:
      5. An undefinable act of knowledge must absolutely exist.

      6. This undefinable act of knowledge must be the light of the world. [metaphor]
      7. Christ is the light of the world [metaphor]

      Therefore from 5. and 6:
      8. Christ is this undefinable act of knowledge.

      Finally from 5. and 8:
      9. Christ must absolutely exist.

      Excuse me if I am not entirely convinced.

  88. manselton
    Posted March 25, 2012 at 4:48 pm | Permalink

    As a theistic evolutionist I am well aware that claiming anything is designed by God immediately precludes it from being understood by science. No one can know the mind of God and if you try to then you end up telling God what he thinks. All fundamentalists do that, sooner or later. When a religious person engages the pursuit of truth through scientific endeavour then the first thing they must do is put God aside. It must be about the science and not about God.

    Science is systematic, formal knowledge. The scientific question is, “How did this phenomenon come to be just as it is?” When that is explained science is complete. The phenomenon is a perceptual experience which forms the point of departure to engage thinking. All knowledge is a combination of perception and thought. The phenomenon lends itself as a stimulus, provided that the person who sees it feels a certain sense of dissatisfaction with the phenomenon just as it is. This makes the subjectivity of perception a necessary stage in the process of knowledge. You must go through subjectivity to achieve objectivity.

    Human beings hunger for knowledge. I believe they do so because as individuals they feel estranged from the world and scientific knowledge enhances the value of human life by re-uniting the person with the world in the most satisfying manner. In this day and age science does this far better than religion, at least in my view. In this I-World isolation my relationship to God can really only be personal to me. Thus I am not an evangelist and – you were right – I ought to keep it to myself or at least only speak of it where the company is appropriate. My apologies for being so blasé.

    • Posted March 26, 2012 at 12:55 am | Permalink

      If only everybody would admit that their faith is NOT based on evidence or reason — that it’s merely a personal conviction or personal preference, then there would be nothing (religious) to fight about any more. There would be nothing to disagree about.

      Denial is at the root of (monotheistic) apologia: denial of the fact that there is no evidence or logical basis for faith. Faith is what it is: belief without the benefit of reason.

      And denial always has a way of coming back to bite us on the butt. Anybody can confirm this fact by watching CNN at any hour of the day or night.

      The majority of the world is certain of things they can’t possibly know anything about. It’s a major problem for human progress.

      • Posted March 26, 2012 at 12:59 am | Permalink

        Oops, I meant to reply to main article, not to manselton!

    • Steve
      Posted March 26, 2012 at 6:56 am | Permalink

      Human beings hunger for knowledge. I believe they do so because as individuals they feel estranged from the world and scientific knowledge enhances the value of human life by re-uniting the person with the world in the most satisfying manner.

      …or they hunger for knowledge so that they might reap a greater amount of future happiness.

      Re-uniting? Assumes a fact not in evidence.


2 Trackbacks/Pingbacks

  1. [...] Polkinghorne’s empirical evidence for god: math and a comprehensible universe « Why Evolution Is …. Share this:TwitterFacebookLike this:LikeBe the first to like this post. [...]

  2. [...] we have the “sophisticated theologians”; those who point out that mathematics is very useful (e. g., the mathematical formalism for quantum mechanics doesn’t explain, but it does [...]

Follow

Get every new post delivered to your Inbox.

Join 29,481 other followers

%d bloggers like this: