John Allen Paulos, in an interesting essay on the co-option of mathematics into religious apologia, makes a useful explanation. To counter the idea that the elegance of mathematics is a reflection of the divine, he suggests otherwise — it is a reflection of the natural world.
The universe acts on us, we adapt to it, and the notions that we develop as a result, including the mathematical ones, are in a sense taught us by the universe. That great bugbear of creationists, evolution has selected those of our ancestors (both human and not) whose behavior and thought are consistent with the workings of the universe. The usefulness of mathematics is thus not so unreasonable.
Sad to say, the comments to the article are a bit depressing: the creationists descend and start yammering about transitional fossils and mangling Gould and that sort of thing, the usual foolishness we expect from them. It deserves better.
It’s just like usual, an inability to think “outside of the box”, so to speak. Christopher Hitchens did a wonderful job of illustrating this point in the very beginning of his new book. His religious teacher said “God made the trees and grass green to be pleasing to our eyes”, while in reality it is our eyes who have adjusted to make the color green pleasing. The same applies to this article. Until you’re able to look at the big picture and see how everything is interwoven amongst each other, it’s almost inevitable that people will attribute it to God.
If advanced mathematics (400 level and above) has taught me anything it’s that math isn’t a reflection of ANYTHING, it is a totally arbitrary made up system with no bearing on reality.
That it can be used to describe and predict the natural world says more for the cleverness of humans at shoehorning anything into anything than it does for the cleverness of our maths. (though to be fair thats what the blurb says too)
Did anyone else see the title of his next book?
Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up
Looks delightful.
This reminds me of a great quote:
“Science is like a differential equation.
Religion is a boundary condition.”
Alan Turing
Besides, because of mathematics, we can ask an interesting question:
If God is infinite, then what is his cardinality? What sized infinity?
We also love the beauty of the predictability and determinacy of math. But. . . what about chaos theory? And what about unsolvable equations such as x + sin(x) = y; you cannot solve for x analytically. What was God thinking?
Sorry: Math Nerd Alert!
Several years ago, one of my math students reported to me after reading Paulos’s Innumeracy that he was upset by the book’s mockery of the story of Noah’s Ark. After all, if God ordained that something be so, how could a mere mortal gainsay it? With God, you see, logic is superfluous.
I think similar reasoning was used by the Australian guy who says he used to be an atheist until his mother died. Her suffering revealed to him that pain is purification, so he trooped off to Egypt to live as a hermit in the desert. Overreaction much? Globe Trekker‘s Megan McCormick visited his cave for a bit of local color in her episode devoted to Egypt. I transcribed some of the hermit’s remarks: [Link]
The comments were more than 100-odd; so I did not bother… But I so wanted to innocuously ask the most faithful of the commenters, if Math was indeed a gift from God, which God they believed handed it into the human beings… The Judeo-Christian God that they recognize? Zeus, Poseidon, Aphrodite, of earlier times?
History of Modern Mathematics clearly states that modern European mathematical system is indebted to Hindu numeral system (popularized in the West through Arabic literature), with use of decimals (in use by the inhabitants of the Indus valley civilization by 3000 BC), use of positional decimal (by the 3rd century BC), and documented use of Zero (by 5th century AD). So would they agree that Maths was given to humans by the Hindu God/Brahman/Ishwar etc.? Did the Hindu God sneak it by while the Judeo-Christian God was sleeping, perhaps? Hmmm…
By extension, shouldn’t these people also agree on the existence of a Hindu God apart from a Judeo-Christian God? Oh, wait! That is blasphemy…
It is like a ball of wool (yes, the same wool that religion pulls over people’s eyes); should you pull one string, gradually all of it come unraveled. Why can’t these people see that? Why can’t they understand that science and scientific thoughts, because they rely on evidence, provide a much firmer and stabler foundation for human activities compared to a hodge-podge of fairy tales, dicta and dogma, collectively called religion?
The comments were depressing, especially considering that the Paulos’s column was posted at ABC News. The prevalence of so many fixed minded religious types says worlds about who watches/reads ABC News. Disney (the owner’s of ABC) should worry, maybe. On the other hand fundies are born suckers for advertising con games, so maybe ABC knows who their audience is. Paulos, however, should be more widely read. Perhaps, SEED could get him to create a blog to be added to their ScienceBlogs. On the other hand, with his posting columns at ABC, maybe a few doubting fundies will see the light–but don’t hold your breath.
If math is a gift from god, how the hell could be mess up something as easy as pi?
The sad thing about this is that it’s very hard for a mathematician to disagree. Math does “transcend” logistics and systems and is a search for “ideal” and “perfect” abstraction. If one were to believe in God or were to allow the word “divine” to have any meaning then obviously math is a “gift from God” (isn’t everything) and expresses the “nature of God” and is “divine” but this doesn’t mean anything more than hyperbole.
I’m not sure how upset to get about it though. Mathematics is the abstraction of reason (and thus important to all who value reason). To Xian’s and creationists God and the divine are the most important tenet to all thought. It’d be inevitable that Xian’s and creationists would tie them together.
Nice to see Pollos, whom I’ve always pegged as a straightforward abstract and popular mathematician and not a natural or qualntitative scientist or philosopher to not take the semantics stand, though. I’ve always figured as a mathematician if one chooses to believe in God or the divine one must fit it into what is and as the “natural world” is what is then “the supernatural” world exists to us as an abstraction and one can interpret an abstraction in any consistant way one pleases and it’d be equivalently meaningful. Hence believing “the supernatural” exist on a “spiritual plane” or exist “somewhere” as platonic ideals or don’t exist would all be equivalent. It’s hard for me to realize that’s not obvious and creationist and fundimentalists insist that “the supernatural” exist in some natural way (sorta defeats the purpose of being “supernatural”, doesn’t it). Well, good for Polos!
Cody (#3):
It’s interesting that Paulos has chosen to write such a book — he seems to have grown a great deal less reluctant over the years.
Form Amazon:
… organizing his book into twelve chapters that refute the twelve arguments most often put forward for believing in God’s existence…” … On the playlist are the firstcause argument, the argument from design, the ontological argument, arguments from faith and biblical codes, the argument from the anthropic principle, the moral universality argument, and others.”
What are the other six?
My counter-args (which are not unique)
1)firstcause: circular reasoning
2)design: is countered by natural selection which fits the evidence better.
3)ontological: begs the question and thus faulty logic
4)faith and biblical codes: simply not convincing.
5)anthropic: um, doesn’t the anthropic principal *weaken* the idea of uniqueness.
6)moral universality:not convincing
What are the other six?
In regard to comment 2, if mathematics is totally arbitrary and has no bearing on reality, then mathematical decriptions of gravity can be safely ignored.
If advanced mathematics (400 level and above) has taught me anything it’s that math isn’t a reflection of ANYTHING, it is a totally arbitrary made up system with no bearing on reality.
Posted by: Onymous | September 3, 2007 4:00 PM
Ok, so Pi*R^2 is purely arbitrary. Substitute 1/0 (being arbitrary here) in your 400 level and above math and see how far that gets you.
Salt, that’s like saying ‘substitute 3000 for the number of feet in a mile and see how far that gets you’. We cherry-pick that relationship out of the enormous wilderness of arrangements of terms because it’s useful – or do you think the counterintuitive numerical value of pi was somehow preordained?
– or do you think the counterintuitive numerical value of pi was somehow preordained?
Posted by: Schwa | September 3, 2007 5:34 PM
Could a circle be anything but a circle, no matter what name you give it?
Pi is anything but arbitrary.
RE: picture of a WW I soldier circa 1912
His belt buckle is brass with a silver badge in the center, unchanged since 1847. The badge has the Prussian king’s crown in the center and the Prussian State Motto, Gott Mit Uns (God is With Us), surrounding the crown.
http://www.worldwar1.com/sfgeruni.htm
So if the atheists are right, the picture is obviously a Prussian WWI Nazi, and this proves that Hitler was a Christian!
(MORONS)
Firstly, you are posting on the wrong thread.
Secondly, your link doesn’t lead to any photo of a belt buckle.
And thirdly, let me doubt that in WW I the belt buckle already had the swastika on it.
While I am at it, German does not have separate capitalization rules for headlines. Nouns always start with a capital letter, other words never do unless they start a sentence or form part of a proper name. Thus: Gott mit uns.
A very odd kind of Christian, admittedly. And Himmler and Bormann weren’t Christians at all. They weren’t atheists either, however!
Pot, kettle.
Firstly, you are posting on the wrong thread.
Secondly, your link doesn’t lead to any photo of a belt buckle.
And thirdly, let me doubt that in WW I the belt buckle already had the swastika on it.
While I am at it, German does not have separate capitalization rules for headlines. Nouns always start with a capital letter, other words never do unless they start a sentence or form part of a proper name. Thus: Gott mit uns.
A very odd kind of Christian, admittedly. And Himmler and Bormann weren’t Christians at all. They weren’t atheists either, however!
Pot, kettle.
The value of PI is only counter-intuitive to a non-mathematician. The probability of a real number chosen at random (if that were possible) being non-transcendental is zero.
His religious teacher said “God made the trees and grass green to be pleasing to our eyes”, while in reality it is our eyes who have adjusted to make the color green pleasing.
Unless, of course, you’re color blind like my best friend. Then it all looks kind of brown. Which kind of messes up the “God Did It” argument.
I (think I) disagree with PZ here.
e^(i2pi)+1=0 (for instance) is a mathematical truth that is quite independent of the universe you happen to have found yourself in. Mathematics exists as a Platonic reality which is independent of physical law.
Of course, there’s no intelligent designer who created these truths, but they don’t even have a naturalistic origin- that’s because mathematical truths don’t have any origin at all- they are eternal and outside of time.
I apologize for my ignorance about extraterrestrial education systems, but what does the 400 level and above mean ? 400 + semesters of algebra or what ?
.
Facts ARE hard, aren’t they…
They all are wearing the brass and silver belt buckle, with the Wurtemburg king’s crown surrounded by the state motto, Furchtlos und Treu (Fearless and True).
And this…
His belt buckle is iron and painted field grey. The badge in the center of the buckle has the Bavarian king’s crown surrounded by the Bavarian State motto, In Treue Fest (In Loyalty Steadfast).
And, of course, the PRUSSIAN belt buckle…
The badge has the Prussian king’s crown in the center and the Prussian State Motto, Gott Mit Uns (God is With Us),…
One thing I’ve noticed about most Americans is that they really have NO FRACKING CLUE to the pre-WWI organization of Germany. And while I don’t know enough to talk too much about it, I know that Germany was hardly a monolithic country and wasn’t ‘united’ (and I use that world loosely) until the mid-to-late 1800’s. Hence the multiplicity of belt buckles.
First they came for the biologists, but I said nothing, because I was not a biologist.
Then they came for the mathematicians …
Christian, could you rewrite your equation for an universe where the metric is d=abs(X-x)+abs(Y-y)+abs(Z-z) instead of the usual euclidean metric ?
In most the colleges I’ve been to, it works something like this:
100 level courses are typically introductory, freshman level courses. Accounting 101, Math 150 (calculus where I came from), etc.
200 level courses are typically sophmore courses and usually require a prerequisite course. English 235 (Creative Writing) would require English 101 (Composition).
300 level courses mean you’ve gotten to your junior year and we’re starting to separate the wheat from the chaff. In my old school, we’d already eliminated half the prospective candidates from my major as being “not good enough.” Intermediate Accounting I & II (310 & 311), Cost Accounting (330), etc. were the offerings. To take these courses you’d need credit in various 100 & 200 level courses.
400 level courses mean you’ve (theoretically) mastered the introductory and intermediate courses and are now focusing on more advanced and/or narrow, subjects. Auditing. Individual taxation. Advanced accounting. Generally tough stuff that builds on the 300 level course work. These are taken by second/third semester juniors and seniors in college.
500 level courses are advanced Senior courses for which you can also get credit towards a Master’s degree. Advanced Cost Accounting. Introduction to Corporate, Estate and Gift taxation. Some others.
600 level and higher are graduate courses.
Thx, Moses, for introduction to american college education, (well it really seems like extraterrestrial compared to our country ) :-),
So, this omynous guy merely says he’s got a dull job, and, most probably lack of intellectual curiosity ;-)
[sigh of relief] I started to be afraid that the usefulness of math is only an illusion caused by my lack of education :-)
Interestingly enough, I am told that there are quite a few people in mathematics whose personal philosophy of same is some form of Platonic idealism: that is, they think that the math objects they work with are objects in the world of forms, in a sense more real than the objects we see in nature, and that all empirical investigation reveals but a shadow of this underlying mathematical reality. I have read that this is a surprisingly popular position even amongst those mathematicians who describe themselves as non-believers.
I’d be interested to learn the opinion/experience of folk like Jason Rosenhouse or Blake Stacey on stuff like this, since they have a lot more math expertise than most folk here, and (definitely) including yours truly…SH
I’m awaiting the new book “Constipated Mathematician, (working things out with a pencil)” I hear it’s sponsored by Sundance Prunes ” Sundance prunes in the morning will get you going all day”
But I digress. If math is divined from Jebus H. Kay-Ricest, how much I have such a hard time with it?
No, no, no – the Divine is an expression of mathematics, not the other way around! :-)
Less witchily – is it true that Euler’s Identity must be so for every possible universe? I was reading some amazing stuff about topoi in New Scientist which, IIUC, meant you could create consistent systems of logic and mathematics different to our own.
I have a question about Mathematical Jesus. Does he have the same blond hair and piercing blue eyes as my old baby-sitters’ “Hologram Jesus” that so freaked me out as a small child? And does he stare at you creepily and follow you around the room with his eyes…
Re #18:
But that’s exactly why one would imagine that there is no divine origin for mathematics. Pi (or e, or a lot of the numbers that describe real objects) are essentially random numbers. It’s only counterintuitive that pi is irrational if you think that God would have put us in a universe where all the math was really easy.
on a euclidean plane sure but in spherical or hyperbolic geometries Pi*R^2 is just some number, or on a torus or a dumbell or any of the more “interesting” topologies.
I’m not trying to make some science destroying statement, I’m an astrophysicist by trade. I’m just saying that once you’ve spent a semester or two proving things with out division because division isn’t really defined in the set of integers (that was a 300 level class) you kinda realize that math was designed to be consistent and logical and useful with itself. That it works with nature is mostly because we made it work with nature (as the blurb says we cherry picked the bits of math that are applicable to the natural world) not because either math or nature are really designed with each other in mind.
dull job yes (it involves babysitting a telescope for hours on end watching 1000’s of identical images get taken, followed, occasionally, by a brief scurry of action if we think we found something), not sure what the lack of intellectual curiosity crack means.
by 400 level and above (really though there were a couple 300 levels that did it too) i just meant math classes high enough up that no one actually needs them to graduate unless they want math degree.
yeah you don’t really run into the arbitrariness when partial differential equations (the requirement for my B.S. in physics) is the highest math you need.
I suppose next we’ll see a entire encyclopedia-documenting
the godly influence on every single last thing from A to Z-
appear in a Memphis library.
I wonder what Jorge Luis Borges would think of the creationists’
massive rewriting of reality….
(read Tlon, Uqbar, Orbis Tertius by JLB if you haven’t
already…pretty unnerving.)
I am curious about the textbook cited on the first page of the article; and I wonder whether the authors credit a Muslim philosopher with the “invention” of algebra and algorithms.
“But that’s exactly why one would imagine that there is no divine origin for mathematics. Pi (or e, or a lot of the numbers that describe real objects) are essentially random numbers.”
I’m an atheist myself, but I don’t believe these values offer evidence one way or the other. It’s no more valid than the bible-thumpers trying to claim that math points to the existence of god.
I think what Onymous was trying to say (and saying it poorly and offensively) is that Mathematics is not dependant upon nature and is completely self referent. It was believed and is believed by most non-mathematicians that math is an abstract generalisation of basic laws of reality. (2 + 2 = 4 means whenever you add two rocks together you will have four rocks). ’tain’t so. Mathematics takes on it’s own relevance and meaning which often has nothing to do with any nature or reality. Often we assume, well, we’re just establishing for possible future cases of reality, or fine tuning the reality behind reality. But ’tain’t so.
Now, it’s interesting to wonder that if our math is independant of reality where the heck does it “come from”. The answer has to be our hard-wiring of our brains which makes one wonder would math be different in a different brain. Likewise, we can imagine reality being different but is it possible for *math* to be different..
Those questions are very uncomfortably close to arguments for God as a transcendant that I’m sure that’s why they appeal to Theologians. Oddly enough, that thought never occured to me until two years ago. I was once put upon to answer why I thought it was valid to “believe” in mathematics and declare it right, yet declare a “belief” in God as possibly wrong. I was thrown through a loop, but I ultimately figure mathematics was a platonic ideal and platonic ideals only exist as language constructs. Mathematics, as opposed to simpler ideals such as “table”, have very determinable and rules during contemplating. I believe the term God exists as a platonic ideal (as do unicorn, evil, indescribable hat, and square circle) but I find very little of interest or practicality in it.
I don’t know what you mean by an “equation for a universe”.
But such a metric is perfectly consistant and easily imaginable. It’s the “taxi-cab” metric. If we apply it to the infinite points of R^3, two points would define an infinite number of lines, not merely one. All triangles would be isoceles (I think). And so on.
?
?
When we humans finally ask the right questions physics will supply the answers. The language of physics is math.
Re the question in #38: One says that division isn’t really defined in the set of integers because most quotients of integers aren’t themselves integers (2/3, for example, is not an integer). The operation of division on integers prompts us to define the rational numbers (ratios of integers). As long as we avoid the dreaded (and meaningless) division by zero, division works very nicely on the set of rational numbers.
God is a rounding error. That’s why nobody’s seen any trace of him, except down in some least-significant bits someplace.
Not a mathematician, but I used to have a bumper sticker that said, “I brake for lattice animals.”
The book “Conversations on Mind, Matter, and Mathematics” suggests that our understanding of mathematics might not be so much a “reflection of the natural world” as a result of the physical structure of the brain. It is an interesting read, the book consists of conversations between a neurobiologist and a mathematician.
Well, I’m not really trying to be pendantic but: division isn’t defined on the integers because 0 is an integer and dividing by 0 is dreaded. Division *is* defined on positive integers or non-zero integers. But it’s not “closed” on such. 6 div 2 = 3 but 6 div 4 has no positive integer solution. To close it we introduce the rationals. But then it’s not closed under square roots and so on. I’m avoiding the whole 0 business. The whole thing really depends of whether you take the algebraist approach or the analyists. Some folks get upset that you have to figure out how to define 6/4 to be the same as 3/2. Other people don’t. Anyway “closing” the integers on division (which we “must” because division is the inverse of multiplication and avoiding the dividing by 0 by making exceptions for the additive inverse) give the rationals. But now we have the whole “imcompleteness” issue which we have to introduce the irrationals for. “Completing” (to get the reals) rather surprisingly closes positive roots. x^2 = 2 doesn’t have a rational answer. But it has a real answer. Even though solving positive roots had nothing to do with why we complete. Lots of folks are happy here. Some folks figure we need to close so x^2 + 1 = 0 will have a solution. Closing here we get the Complex numbers. The complex numbers are closed *AND* complete. Cool. But they aren’t ordered. Is sqroot(-1) bigger or less than 1?
For my senior seminor I gave a talk on the p-adic metric. Addition, subtraction, multiplication, and division and closing gave us the rationals. But, if I remember right (it *was* 23 years ago), Q *was* complete with this new metric. But it wasn’t closed under roots. Closing under roots it became incomplete. Completing it stopped being closed. Closing it it became incomplete again!
Weird. But fun.
but they aren’t random or arbitrary. In fact pi is a probability of random integers being relatively prime. The thing is, wierd as this result is, it’s provable and it *has* to be so. Those who want to praise god’s work everywhere will find the extraordinary and evidence of “his infinite wisdom” but others will say that it’s provable so it *had* to be so. Even if there is an adjustable guage for gravitational constant or or magnetic force to set the universe there *cant* be any such gauge to set pi to or to set the probability of relative primeness to. So we *could* say God had a choice when it came to creating man but he didn’t have a choice when it came to creating pi. So God’s not omnipotent. This is utterly meaningless and pointless though.
I think you meant 6/π2. Probability is usually defined as being in the range [0,1], π is too big.
I’ve no idea what TUT’s trying to say either. The universe we live in has a very complicated metric, but e^(2 pi i) + 1 = 0 all the same. pi is the ratio of the circumference to the diameter of a circle on a Euclidean plane. The fact that we don’t actually live in a Euclidean space doesn’t change the definition of pi.
(Btw, I think TUT’s metric is also known as the Manhattan Metric, for obvious reasons.)
Paulos wouldn’t have had much to write about had Pharyngula not exposed that San Antonio school’s math program to the wider world a couple of weeks back.
In contrast to your (humorous) comments about bloggers not having a life, some blogs make a difference. Pharyngula is one of those, maybe the best among them.
“The number i is evidence that much real progress can result from the positing of imaginary entities. Theologians who have built elaborate systems on much flimsier analogies should perhaps take heart.”
My favorite John Allen Paulos quote.
on a euclidean plane sure but in spherical or hyperbolic geometries Pi*R^2 is just some number, or on a torus or a dumbell or any of the more “interesting” topologies.
Even in non-Euclidean geometries, as long as the geometry is differentiable on some open set then on that set Pi should still be “special” as the limit of A(R)/R^2 as R approaches 0, right? Is there a differential geometer in the audience?
I’m going to pull my dead horse out of the glue factory to flog it a few more times.
Pointing to the evident consistency of mathematics as an indicator that it has some kind of existence independent of thought is a non sequitur. Likewise, the seeming amenability of reality to mathematical description says nothing about the existence of mathematics in an objective sense. Mathematics possesses its descriptive abilities by construction, and its evident consistency by careful axiomatization of those same intuitive generalizations. All of its seemingly mystical properties have their origins in forgetting that mathematics is a somewhat arbitrary construction. In any case, playing the latter day Platonist plays into the hands of the apologists who are trying to co-opt mathematics for their piss poor philosophy.
Here are some good antidotes to MagicManDunnit Mathematics:
“Outlines of a Formalist Philosophy of Mathematics” by Haskell B. Curry (out of print, in most libraries).
“Social Constructivism as a Philosophy of Mathematics” by Paul Ernest (in print, not in most libraries).
“Wittgenstein’s Lectures on the Foundations of Mathematics” edited by Cora Diamond (features several arguments between Wittgenstein and Turing on mathematical formalism).
“Philosophy of Mathematics: Selected Readings” edited by Hilary Putnam and Paul Benacerraf (see, especially, “Truth by Convention” by Quine).
I think that might be a chicken-egg kind of thing. The brain possesses the structure that it does so that it can provide a workable model of the goings on outside the skull. The neurology would certainly indicate that the manner in which we go about doing mathematics has a lot to do with the structure of the brain, but the cognitive functions of the brain are there, after all, to provide a “good enough” description of reality.
It would be interesting to meet some little green men, and see what they think about math.
When I get tired of the extraordinarily sterile debate between theists and atheists (when I participate, I’m on the latter side) I try to come up with some notion of God that is not incompatible with science, not stupid, and not trivial. I usually arrive at some sort of quasimathematical formulation. See here for more (if there’s anything original there, it’s at the end).
To Dustin, thanks for providing those phil of math references, but it’s not like the Platonist/formalist debate is settled (is anything every settled in philosophy) and there are plenty of reasons to prefer Platonism, without taking it as any sort of support for MagicMan theism.
Dustin, your scholarly post has one serious problem, according to my lights.
The critical phrase, “intuitive generalizations, ” is pregnant with Platonism.
My “intuition” is that the earth is flat.