The (Unsurprising) Effectiveness of Mathematics in the Natural Sciences

“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.” – Pope Benedict XVI

In 1960, physicist and Nobel laureate Eugene Wigner published a well-known article entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences“. In it, he explored the mind-boggling usefulness of mathematics in describing and further revealing the physical universe. As an example, he questioned why tools such as complex numbers – foreign as they are to our intuition and everyday experiences – are nonetheless a product of human thought necessary for the formulation of the laws of quantum mechanics. Why is it that elements of mathematics can so perfectly describe seemingly unrelated phenomena, such as the ratio of a circle’s circumference to its diameter (pi) appearing in the Gaussian distribution function used to study population trends? Furthermore, why is it that the human mind is so astonishingly capable of recognizing the language of mathematics and using it to advance our knowledge of the physical world?

Of course, these questions inevitably lead to others. Why is the universe so ordered in the first place? Where did this order come from, if anywhere? This apparent “fine-tuning” of the universe (particularly with respect to the existence of intelligent life) has numerous possible explanations. I’m going to briefly describe three of the most prevalent, but be aware that some of these have sub-variations, and that this isn’t an entirely comprehensive list.

1. Perhaps there are actually multiple universes (or even infinite universes), and we happen to exist in a universe with physical properties that allow us to exist.

2. Perhaps the universe isn’t actually as ordered as it seems.

3. Perhaps the universe “just is”. Maybe some day we’ll know why; maybe we never will.

Of course, the reader will have doubtlessly recognized that all of these explanations avoid what is probably the simplest and most intuitive response: that our seemingly fine-tuned universe was indeed finely tuned by a Fine Tuner. Although many of today’s scientists balk at the notion that the universe may have been designed, it is worth pointing out that this position is no less scientifically defensible than, say, the idea of multiple universes. Neither can be empirically tested. Both can only be accepted on faith. Design is also less problematic, practically speaking, than the idea that the universe’s apparent order is illusory, and it is considerably more satisfying than simply accepting the apparent order as an unknowable brute fact.

All of this brings me back to the Pope Benedict quote that I opened with. If the intellectual structure of the human mind truly does coincide with the objective structure of the universe – as it undoubtedly seems to – then what implications does this carry for the Christian? This is the kind of question that obviously cannot be answered scientifically, but that makes it no less fascinating. I can’t pretend to actually know all the answers, of course, but I do have a few ideas.

Genesis 1:26-28 states that God created mankind in his image, so that man could subdue the earth and rule over its creatures. I think when most people hear the phrase “in God’s image”, they tend to think strictly in terms of physical appearance. They imagine a guy with a flowing white beard breathing life into Adam. But what quality is it, really, that enables man to subdue the earth? Although our physical stature is certainly impressive, it’s clear that we’re NOT the largest, strongest, or fastest creatures on this planet. What makes us unique, rather, is our minds. When we’re told that God created mankind in his image, does it not seem likely that this applies to our intellect as much as it applies to our physical appearance?

Consider what this means. If the God who designed the universe and set in place its mathematical properties also created us in his image, it would be reasonable to expect the human mind to be fluent in the language of mathematics. It would hardly be surprising to learn that we’re inhabiting a finely-tuned universe, or to find that our language of mathematics is so powerfully capable of uncovering the physical laws that govern it. For the Christian, the problems that have so long confounded physicists and philosophers aren’t really problems at all. Man’s intelligence is simply reflecting a small shadow of God’s intelligence.

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31 thoughts on “The (Unsurprising) Effectiveness of Mathematics in the Natural Sciences

  1. Pingback: Shared: Effectiveness of Mathematics « Cloak and Sword

  2. Reminds me of the work Reasons to Believe makes, particularly a book I read from one of their own called “Who’s Afraid of the Multiverse” by Jeff Zweerink. I think the human experience as a whole is a beautiful depiction of a Creator.

  3. Wow man.. I totally didn’t see you ending with that conclusion.. I was seriously getting excited as I read haha.. I actually wrote a blog (re-posted from my other blog) a while back called the Mind Of God which kind of goes along with what your saying. It’s based in John 1-18.. very cool stuff man… Good post!

  4. Amen. I am not a scientist, not even further education but always concidered how strange the idea that “time and chance” has to produce anything complex, unique and designed, no matter how many millions of years they allow it.
    To me, a layman in these things, to benefit from “time and chance” there must be a built in mechanism of memory to learn, select, and copy in the simplest unit which speaks of something more complex, ordered and controlled, following rules and laws otherwise “time and chance” would produce chaos and a continuing mulitude of strange aberrations and entirely new things. How do non Christian scientists explain the lack of these random variations that should be so prevalent?

  5. Extremely well written and thought out! Thanks for stopping by at LightWriters (5wise.wordpress.com) and I’ll be following your blog with interest! God bless!

  6. I like the way you have worded the idea of intelligent design: “that our seemingly fine-tuned universe was indeed finely tuned by a Fine Tuner.” I agree that this is the simplest and most intuitive response, even if it is empirically unproveable. Peace, Linda

  7. Great read!~

    I remember when my cousin had to tutor me in mathematics as a young teen and he said, Sherry, numbers don’t lie. They will always add up right. I’ve loved math ever since and even, in my senior year of high school, got so far ahead of everyone that my teacher gave me his college calculus book to study.

    Anyway, I’m glad our Fine Tuner was perfect in mathematics because our universe would be unfit for life if it were off just a hairsbreadth. 😯

    Loved what you said here:
    Man’s intelligence is simply reflecting a small shadow of God’s intelligence.

  8. The believer in me “wants” to agree with your statements. The doubter in me recognizes that given sufficient time and opportunity, we can/will inevitably get the universe we exist in. An understanding of math/statistics/probability does not negate my belief in a “tuner”. Science does not need a defense; neither does a creator. One seeks to explain what and how; the other explains the why.

  9. Math has always been above my brain, so that in itself means that anyone (including God!) who knows or uses mathematics must of necessity be a Higher Intelligence than me…easily.

    Very nice post, and thank you for using the quote from Pope Benedict too! God bless.

  10. Matt, thanks for visiting my blog and for ‘liking’ my adventures. I found this post of yours very interesting and as you’re about to find out, I love this sort of discussion…

    I know popes are supposed to be infallible, but I think Benedict XVI has slipped up here.

    From the point of view of humans, yes indeed, the universe appears unusually perfectly designed for our survival. Even the Earth itself has seldom in its 5 billion year history been suitable for warm-blooded mammals to feed, reproduce, watch reality TV and kick footballs on.

    I know about the Goldilocks Enigma – Earth is not too hot, not too cold, with breathable air and drinkable water and gravity exerts precisely the correct force to allow atoms to come together to form galaxies, stars and bloggers.

    However, it is a leap of faith to jump from this to the conclusion that it has been fine-tuned for our human benefit by a divine Fine Tuner, whose mind is assumed to work rather like a human brain, or at least like the brain of a clever human mathematician.

    A bacterium (let’s call him Jim) that lives under my tongue could equally believe that the universe has been constructed precisely for his benefit. He doesn’t need to do any maths, because my saliva is just acidic enough to break down food into protein for Jim to eat, but not so acidic that it will dissolve him.

    There’s enough nourishment in my mouth for millions of Jim’s mates to get along there quite happily. He toils not, neither does he spin, he just hangs out. The universe is perfect for Jim, and by Mr Ratzinger’s reasoning, could possibly have been created with Jim in mind. He obviously belongs to God’s chosen species and lives on God’s chosen planet in God’s chosen mouth.

    Sure he’s lucky to have me hosting him for the moment, but he may be one of those adaptable bacteria that will jump ship and survive even after my demise. From Jim’s point of view, only a kind, brilliant, all-powerful, bacteria-loving God (who may possibly even look like a wise old germ with a beard) would have organised things so perfectly on Jim’s behalf. If things were only fractionally different, Jim would be in big trouble.

    I find the idea that the vast universe was fine-tuned simply as a backdrop for one single species, homo sapiens, to live their lives and worship their Maker from one single planet on a remote arm of the Milky Way galaxy to be ridiculously arrogant. Do we really think ourselves so important that this whole thing was designed as a sort of practical exam to find out which of us are suitable entrants to Heaven and who should spend eternity in Hell?
    It was a shock to the church to discover that the Earth was a planet in orbit around the sun, that the stars were also suns, that homo sapiens was an ape in many respects like other animals and not the final pinnacle of creation, but part of a continuum of evolution.

    Heaven was not a place just above the dome of the sky, fixed stars were not fixed, and disease, drought and earthquakes had natural causes and were not necessarily the work of gods punishing people for sins.

    Church leaders were so shocked and worried about the moral decay that might follow if their flock found all this out, that they tried very hard to suppress the information, persecuting and condemning the likes of Galileo and Darwin. Some religions and Christian churches still prefer their adherents to be ignorant and unquestioning.

    Those that accepted the science rushed about finding some other justification for our species’ hitherto unchallenged self-importance; humans had (invisible, undetectable, eternal, unprovable) souls, while bacteria like my friend Jim did not.
    It’s possible that the universe was created purely for the benefit of humankind, that Earth has been specially selected by a divine Fine Tuner, and that we are a uniquely chosen species and the only species in the universe that has ever or will ever exist with the capacity to consider these matters.
    Possible, but unlikely. There are almost certainly an awful lot of Earth-like planets out there and probably far more possible forms of life than the DNA-based one we know about. It is also highly unlikely that we will ever come into contact with them. Space is just too big.
    It is comforting to some to believe that God made Man in his own image. Comforting and convenient. Such thinking justifies Man assuming for himself a privileged position in the natural order, with dominion of the beasts of the field and fowl of the air.

    The evidence suggests far more strongly that Man makes gods in his own image, such images chosen to appeal to a particular culture. The Greeks believed the gods looked like Greek muscle-builders and lived on Mt Olympus, surprise, surprise, in Greece. Hindus give Ganesh an elephant head because Hindus know what elephants look like – wise and powerful. Some Australian Aborigines believe in a large Rainbow Serpent, who created river valleys as he wriggled through them. River valleys look like snake tracks and they’d seen a lot of them. Aborigines were unlikely to believe in an elephant-headed god, even if the Hindus are right about Ganesh.

    The Hebrews predictably created a god who resembles a chief of a Jewish desert tribe. He insists on being worshipped unconditionally, enjoys having animals sacrificed to him, divides sheep from goats, plays favourites, smites his enemies with plagues and famines, supports his chosen people (the Israelites, who else?) in battle, approves of them slaughtering and enslaving those they conquer, and includes wives next to asses and oxen as property items you shouldn’t steal or covet from your neighbour.

    When he really gets mad, he sends a flood to drown everyone except a few people who’ve sucked up to him.

    He sounds a bit like Saddam Hussein or Gaddafi. We can’t blame the Hebrews for inventing such a partisan and capricious god – that’s what they thought strong leadership involved, and such a god would be useful to have on your side. The best we can say about him is he probably doesn’t exist, any more than Zeus or Ganesh or the Rainbow Serpent do.
    Of course, I may be quite wrong. People claim to talk to him, and believe he intervenes on their behalf, temporarily suspending the laws of nature and working miracles. At other times he seems to ignore the prayers of even the most devout, worthy, loving, sincere, deserving people and they say only that he’s ‘working in mysterious ways’.
    It really comes down, as you say, to faith, the belief in something which cannot be supported by testable evidence. Multiverses or intelligent designers are possibilities, but only because anything that can’t be investigated and disproved is possible, like Bertrand Russell’s celestial teapot.

    After that it’s a matter of taste. Would you prefer some divine dictator to be in charge, occasionally sticking a heavenly finger in to stir the pot on your behalf if you’ve been good? Or if you’ve been bad, sending a junior family member as a scapegoat to take the rap on your behalf?

    Or would you prefer to have more independence, and understand that you are responsible for your own actions, and that you’d better make the most of life here and now, because it’s the best and only one you’re going to get?

    I have no way of knowing whether I will meet a Fine Tuner after I die, but I suspect it is very unlikely. My death will be the gain of bacteria like Jim and I wish them much pleasure as they absorb my protein. The thought does not depress me – I enjoy being useful.

    I can hope to achieve a sort of immortality insofar as what I do in my life will influence those around me, who in turn will pass that on. I don’t believe my parents are in heaven, but I know they live on in me. I hear their voices whenever I begin a lecture to my children with, ‘When I was your age…’

    I’m not uniquely privileged. I’m not great at maths, though I do take an interest in cosmological matters. I wish the Fine Tuner should have done a better job on my brain to help me get my head around curved space-time, black holes, dark matter and Higgs’ boson.

    These seem to be rather vital parts of the objective structure of the universe and the Fine Tuner didn’t equip most of us poor homo sapiens with the intellectual structure to coincide with them. I forgive him this oversight. I’m only human.

    So I’ll try to use my time on this remote planet to form the best relationships I can with my fellow human beings, those I know and those I don’t know, not because I hope for heavenly reward, but because it’s fun and it makes us all happy.

    I’ll try to be humble about the importance of my fleeting existence in time and space. I’ll try to care for the Earth and the fantastic range of life it contains, all of which can claim as much right to exist as I or Mr Ratzinger do. I’ve heard that 99.99% (or some such figure) of all species that ever existed on Earth are now extinct, and I’ll try to slow that process down.
    I’ll take pleasure in being a small part of whole big wonderful universe, and spare a passing thought for bacteria like Jim when I spit him and a few million of his colleagues into the gutter.

    Well, if you’re still there, Matt, thanks for reading, I enjoyed writing it, and happy travels. All the best to you!

  11. Excellent post! I agree that man’s intelligence is only a small reflection of God’s. If we are made in His image, then our physical being, intellectual mind and eternal soul are each reflections of the being, Almighty God.

  12. Very interesting. I’m in agreement with what you have to say– but I tend express things a little bit differently; I don’t think we disagree, but I think we have a slightly different focus. I see numbers as a specific example of the predictabality of the universe. They certainly aren’t less remarkable than other constants in the universe. But I’m not sure that they are more remarkable.
    Consider Pi; it seems to me that it is a remarkable thing that their is a constant ratio between diameter and circumference. In every circle everywhere, it is about 3 times longer to walk around a circle than it is to cut through the middle.
    The fact that we express this ratio as 3.14… doesn’t strike me as particularly interesting. If we had less or more than 10 fingers, most likely we wouldn’t have a base 10 number system, and so this ratio would be expressed with different numbers. I’ve heard some claim that base 1o number systems are uniquely suited for all sorts of convenient short cuts. The utility of this system, I suppose, further implies the existence of a creator (i.e. he knew that base 10 would be a good number system, and this is why he created/evolved us with 10 fingers) but that, I suppose, is a digression.

  13. That was a nice quote from Pope Benedict to open up with. However, I felt as though your post lost its logic somewhere near the middle, as you seemed to muddle up fine tuning arguments which ought to remain distinct.

    Another commentator on this thread, Richard Tulloch, has pulled out a lot of objections, most of them rather trivial – for example, the Pope is not infallible when he isn’t speaking either ex cathedra, or else at least speaks with the authority of his office as Bishop of Rome, but he is certainly isn’t doing that here. The objections about the fine tuning argument are also short of stunning, but he may be excused here precisely because he has followed you in mixing up the arguments. The argument from the fine tuning of the universe for intelligent life is not the same as the argument from the fine tuning of the correspondence between the faculties of the human mind and the world. Obviously Plantinga’s evolutionary argument against Naturalism is a good place to start in trying to explain the difficulty that Pope Benedict, Dinesh D’Souza, along with a plethora of others, see here.

    The fine tuning argument from the original constants and quantities permitting intelligent life is championed magnificently by both Dr. William Lane Craig, and by Dr. Robin Collins – so In the name of brevity I will just direct readers to them.

    Matt, you say: “Although many of today’s scientists balk at the notion that the universe may have been designed, it is worth pointing out that this position is no less scientifically defensible than, say, the idea of multiple universes. Neither can be empirically tested. Both can only be accepted on faith. ”

    I disagree. Not only can design be derived rationally whenever one can infer what we might call specified complexity, but the multiverse hypothesis does little to nothing in offering an escape from sophisticated forms of the fine tuning arguments. In other words, I take it that the design hypothesis is eminently more scientific than the multiverse hypothesis. Moreover, I would like to discourage the use of the word faith – that colloquial use of it demeans it and proposes a meaning which Christian theology has never recognized, and which has only been used in that way since Kierkegaard.

    Very good article though.

  14. “What is it that enables something as small and delicate and adapted to terrestrial life as the human brain to engage with the totality of the cosmos and the silent mathematical tune to which it dances?” – Paul Davies.

  15. An interesting conjecture. I have faith but I’m also a believer of science. I’ve stopped looking for an explanation on how science and religion commingle. I think everyone should just be more open-minded.

  16. Matt, Thank you for the “like” at rivertop rambles blogsite. Lots to think about in your post, but I’m really most interested in your header photo, of the spirit implicit in any careful view of nature. I would probably underscore some of the analysis in Richard Tulloch’s commentary above. The universe is bigger than anyone can possibly get a handle on.

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