Good Math, Bad Math

I was planning on ignoring this one, but tons of readers have been writing
to me about the latest inanity spouting from the keyboard of Discovery
Institute’s flunky, Denise O’Leary.

Here’s what she had to say:

Even though I am not a creationist by any reasonable definition,
I sometimes get pegged as the local gap tooth creationist moron. (But then I
don’t have gaps in my teeth either. Check unretouched photos.)

As the best gap tooth they could come up with, a local TV station interviewed
me about “superstition” the other day.

The issue turned out to be superstition related to numbers. Were they hoping
I’d fall in?

The skinny: Some local people want their house numbers changed because they
feel the current number assignment is “unlucky.”

Look, guys, numbers here are assigned on a strict directional rota. If the
number bugs you so much, move.

Don’t mess up the street directory for everyone else. Paramedics, fire chiefs,
police chiefs, et cetera, might need a directory they can make sense of. You
might be glad for that yourself one day.

Anyway, I didn’t get a chance to say this on the program so I will now: No
numbers are evil or unlucky. All numbers are – in my view – created by God to
march in a strict series or else a discoverable* series, and that is what
makes mathematics possible. And mathematics is evidence for design, not
superstition.

The interview may never have aired. I tend to flub the gap-tooth creationist
moron role, so interviews with me are often not aired.

* I am thinking here of numbers like pi, that just go on and on and never
shut up, but you can work with them anyway.(You just decide where you want
to cut the mike.)

It’s such concentrated stupidity, it’s hard to know quite where to start. So
how about we start at the beginning?

Denise O’Leary claims not to be a creationist by “any reasonable
definition”? Yeesh. No point even trying to argue with that. She’s just playing
the usual ID’ers games with the definition of “creationist”.

Then, very rapidly, we get the usual victimization rant. Poor, poor
Denise. Such an unfortunate soul, so looked down on. I mean, she spews
non-stop nonsense, and all she gets for it is a nice salary, lots of attention,
a publishing contract, and some television interviews. Those IDers sure are
put upon, aren’t they?

Then – shock! – she gets something right. The subject of
the interview was goofy people who want their house numbers changed, because
they think that they got unlucky numbers. Yeah, that’s pretty stupid.
Absolutely.

It brings to mind an interesting story. Back when I was in college, my
family had to move. My parents had taken out a ten-year renegotiable mortgage,
and they couldn’t afford the increased payments while also making the tuition
bills for me and my brother. They ended up selling the house very quickly. But
it was really strange. The people who bought it were Chinese, and they hated
just about everything about the house. They hated the landscaping.
They hated the kitchen. They hated the tiling. They hated the slate foyer.
They hated the windows. They hated the parquet wood floors. They thought it
was too big. Honestly, if there was anything that they actually liked
about the house, I don’t know what it was. But they bought it. Because it
faced in the right direction
, and it was the only house facing exactly
that direction on the market. Their feng shui master had told them that they
must have a house that faced in that direction – that anything else
would bring them terrible luck. So they bought it.

People believe all sorts of strange things. There are all sorts of peculiar
superstitions, about numbers, names, shapes, colors, directions. It’s all silly.
And it’s amazing how many of us still hold on to those odd ideas, or at least
the behaviors that they imply. To get personal, I know perfectly well that
nothing I say is going to cause the world to turn on me and make something
bad happen. But European Jews have a lot of superstitions about drawing attention
to themselves, and I never say things like “Well, things couldn’t
possibly get any worse”, or “Things are so great, I can’t imagine how they could
get better”. Those are both statements that “draw attention”. I know how stupid
it is, but that doesn’t change the feeling I get in the pit of my stomach when
someone says something like that.

So yeah, superstitions like that are silly, and they do deserve to be
mocked. Mine included. But I’ll bet you dollars to donuts that Denise wouldn’t
buy a house where a satanist had performed his phony rituals without getting
it purified by a priest with holy water, and that she wouldn’t see anything
remotely silly about it. She sees her superstitions as legitimate,
but others as mockable.

ANyway, enough of that. Let’s get to the good part.

She says “No numbers are evil or unlucky. All numbers are – in my view -
created by God to march in a strict series or else a discoverable* series, and
that is what makes mathematics possible. And mathematics is evidence for
design, not superstition.”

Oy, oy, oy.

Numbers were not created by a supernatural being. No deity, no matter how
powerful, could have created a universe where numbers didn’t exist, or didn’t
work.

This is a surprisingly difficult and subtle point. But numbers, in some
sense, aren’t real. They’re purely conceptual. There’s no such thing
in the real universe as the number 2. There are plenty of examples of “two
objects”, but the number 2 doesn’t exist. Far worse, there is absolutely no
way of claiming that π really exists. There are no perfect circles in the
universe. And the only sense in which π can possibly exist in the real
universe is as a measurement.

Numbers are an artifact of reasoning. They don’t exist out there in the
void, waiting for someone to find them. They’re a consequence of a simple set
of rules. And those rules must work. There’s no way that God can
change the nature of an abstraction that doesn’t really exist. He could make
it impossible for us to conceive of those rules. But the rules would
still work. Even if there was no universe at all, those
rules could still be said to exist, and therefore, that the numbers still
exist.

It comes down to a deceptively simple question: “What is a number?”. And
there is no single answer to that question! I can define numbers informally,
by counting. I can formalize that a bit, and get two different kinds of
numbers: ordinals and cardinals. I can formalize differently, and get surreal
numbers. Still another way, I can start with different rules, and get Piano
numbers. Or another way, and get computable numbers. I can define real
numbers, complex numbers, vectors, quaternions. Those are all perfectly valid
concepts – and they’re all different. Which one really
defines numbers? All of them. None of them. Take your pick. Numbers are
what you want them to be. They don’t exist outside of your mind. They’re a tool
that we use to understand the universe – but they don’t have any real,
objective reality.

But Denise’s stupidity doesn’t end there. She needs to qualify things – the
numbers “all proceed in a strict series, or else a discoverable series”.

Bzzzt. Wrong.

You can look at that statement in two ways. One way of looking at it – which
I think is the one she meant – is just completely, utterly, wrong. The other way,
which you could reasonably argue is the correct interpretation, is totally
fouled up by that qualification.

Interpretation one:

“The numbers all proceed in a strict series”. My initial reading
of this is that “series” implies a listing or enumeration of one number after
another.

The problem with this is that you can’t put the real numbers into
that kind of series. The real numbers are an uncountable set: you can’t
enumerate the elements of an uncountable set. So they can’t possibly be
put in a series.

You could weasel out of that problem, by saying that the
qualification solves the problem: you can enumerate the rational
numbers: you can put them into a kind of series. Since she explicitly mentions
numbers like π as being exceptions, you could argue that she meant
that the rational numbers could be put into a series, and that the “discoverable
series” qualification was meant to cover the irrational numbers.

Alas, that doesn’t work either. First, from her wording and description, I
really don’t think that when she said the numbers are in a strict series, that
she had in mind an ordering where, for example, 2 comes before 1/3, and
1/3 comes before 1/100. But you can’t enumerate the rationals in
anything like comparison order, which is what I think she was trying
to say.

In addition to that point, I’d say that there’s something seriously wrong
with a definition where the exception covers the overwhelming
majority of cases. Most numbers are irrational – but her phrasing implies that
the irrationals are sort-of strange exceptions.

But I left the worse for last. As I’ve mentioned before, href="http://scienceblogs.com/goodmath/2009/05/you_cant_write_that_number_in.php">most
numbers are undescribable
. You can’t discover them. You can’t
describe them. You can’t name them. You can’t point at them. And yet, by the
definition of real numbers, they must exist. So even forgetting about
the whole ordering issue, the idea of all numbers being discoverable, is just
totally wrong. They’re not. Numbers are much stranger, much less
rational, less intuitively comprehensible, less well-behaved than her naive
understanding.

Interpretation Two

The second interpretation is that “the numbers all proceed in a strict series” is a poorly
phrased way of saying that the real numbers are totally ordered. That is a fact:
given any two distinct real numbers X and Y, either X<Y or Y<X. That’s correct. But if
that’s what she meant, then she blew herself out of the water with the qualification: because
irrational numbers like π are still part of the total ordering of the real numbers.
Pulling them out by that qualifier implies that she doesn’t believe that they’re part of
the series – which in this interpretation means that you can’t always compare them. But even given
two irrational numbers, they’re always comparable. Even the undescribables.

And the qualification still fails exactly the same way it did in case one: most
numbers aren’t discoverable, describable, nameable, identifyable, or enumerable.

So again, she fails miserably.

The takeaway point here is that numbers are both less real, much stranger,
and frankly a whole lot more interesting than Denise O’Leary imagines. As
usual for Creationists (and yes, Denise, you are a creationist!),
she’s taken a simplistic understanding of something, mistaken her simplistic
understanding for a deep comprehension of it, and then argued that on the
basis of its alleged simplicity that it must have been designed by her deity.

Her version of numbers can’t account for undescribable numbers. It can’t
account for much of the beautiful strangeness of numbers. It can’t account for
logical wierdness like Gödel’s incompleteness theorem, which relies on the
logical structure of numbers. It can’t account for some of the magnificent strangeness
that people like Greg Chaitin have studied. As is all too common, she’s so satisfied
with her simplifications that she’s completely missed both the pathology and the beauty
of numbers. It’s sad.

It should be obvious, looking at this blog, that I’m deeply in love with
mathematics. Math is beautiful, and fascinating, and frustrating, and strange.
People like Denise O’Leary try to sap out everything that makes it wonderful
in order to be able to say that they understand it, and that their personal
deity created it. God didn’t create math. Math is a collection of formalisms
that we created from the basic rules of logic – and those rules
must hold, no matter what the universe is like. Because they aren’t
rules about the universe – they’re self-contained rules about concepts that
they describe.

If you’re religious like me, you might believe that there is some deity that
created the Universe. Or you might not. But whether there is a God or not has nothing
to do with whether A∧¬A == false.

Comments

  1. #1 rmp
    October 19, 2009

    This reminds me of your post over a year ago and ‘i’ or ‘j’ or square root of -1. My math prof would go on an extended rant about the term ‘real’ numbers versus ‘imaginary’. He did it so often I swear I’ve almost got it memorized.

    There is no such thing as a Real number he’d insist. Go out in your back yard and dig up a 2. and so on and so on …

  2. #2 Jake
    October 19, 2009

    I thought she meant the integers, myself…

  3. #3 Ben
    October 19, 2009

    re Interpretation 1: Reals are at least orderable; complex numbers are just as valid as reals and they can’t even be ordered.

  4. #4 Aaron
    October 19, 2009

    More like only “Natural” numbers, right? ;)

  5. #5 Susan B.
    October 19, 2009

    Re: rmp

    I figure as long as you’re going to count the real numbers as “numbers” there’s really no reason to leave out the complex numbers. I mean, for most practical purposes you only really need the rationals anyway (you can approximate pi as closely as you want with rationals). If you are going to count the reals, then why stop there? The complex numbers lack only one property that the reals have, which is an ordering that plays nice with the field axioms, and they have many more properties that are highly useful, such as being algebraically closed. Not to mention that the complex numbers are themselves capable of describing certain real-world phenomena.

    Of course, I’m not saying we should stop at the complex numbers either. My general feeling is that if you want to define “numbers”, you need to decide first of all which axioms, structures, or properties you’d like “numbers” to have and go from there.

  6. #6 Mark C. Chu-Carroll
    October 19, 2009

    @2:

    She couldn’t have meant integers: she explicitly mentions π. So she’s got to be including rationals and irrationals.

    But the point is, she thinks that there’s a collection of things called numbers, and that they’ve got some kind of objective existence. But in reality, numbers are what we say they are; and we typically use the word “numbers” to mean lots of different things – and depending on what we mean, we’re talking about totally different things. Are we talking about cardinal numbers? Are we talking about complex numbers? Computable numbers? Surreal numbers? Real numbers? Peano numbers?

  7. #7 Hank
    October 19, 2009

    She’s probably talking about Humpty Dumpty numbers. They have whatever properties the creationist in question wants them to have, no more, no less.

  8. #8 Peter
    October 19, 2009

    Why would you consider someone stupid for not believing your interpretation of a “a surprisingly difficult and subtle point”?

  9. #9 CJ
    October 19, 2009

    I actually feel that the only good evidence for the universe being anything more than the observable facts is the fact that our ape brains have conceived of this system we call ‘mathematics’ which appears to describe the universe extremely well at scales which our brain never evolved to comprehend (conceived without us intentionally trying for this end – the theory of ODEs et cetera predates by far general relativity and quantum theory). That’s not what she said, but it’s in the same kind of vein.

  10. #10 zingle
    October 19, 2009

    I thought the standard line was “God created the natural numbers, all the rest is the work of man.”

    (Not proposed as a contention with anything MCC says, it just seems obligatory to mention it in this discussion.)

  11. #11 Nate
    October 19, 2009

    I think it’s unnecessary to rant at her lack of understanding of number theory; the secondary point she was making (which is far more interesting to rant about) is that she believes that mathematics the concept is necessarily a product of design, not “superstition”.

    Based on your response I think you’d agree with her there, in that mathematics is not defined by the universe, even as it makes use of the same. Math is just a natural consequence of logic, and it is a “designed” construction. Of course Denise thinks that an invisible old man did the designing and intelligent people think that various animal intelligences do the designing (at different levels of proficiency and formalization).

    The universe doesn’t anthropomorphically need a definition of real numbers in order to operate but we humans find that a definition of real numbers is key to understanding the universe and describing its operation.

    Anyway, I’m not sure that Denise was wildly off the wall here. Of course her mathematics is faulty, but she’s not a mathematician and the specifics weren’t crucial to her point.

    As for her claiming she wouldn’t fall into a superstition trap… well, who wants to bet she wouldn’t accept a hip replacement with serial number 666?

  12. #12 eee
    October 19, 2009

    Obviously she doesn’t know what she’s talking about, but as Leopold Kronecker said: “God made the integers; all else is the work of man” (not that I happen to agree, since I don’t believe in God)

  13. #13 Mark C. Chu-Carroll
    October 19, 2009

    @8:

    I thought I was very clear about that.

    She’s got an incredibly shallow and simplistic concept of numbers. But that shallow and simplistic understanding fits what she *wants* to believe – and so she doesn’t look any farther.

    In particular, she wrote her babble it as part of a rant about how people stupid were for attaching superstitious meaning to numbers, while simultaneous attaching her own superstitious meaning to numbers, and proclaiming that her superstitious meaning is a demonstration of how smart she is, compared to the gap-toothed morons.

  14. #14 Doug Little
    October 19, 2009

    I am thinking here of numbers like pi, that just go on and on and never shut up, but you can work with them anyway.(You just decide where you want to cut the mike.)

    Speaking of the number 2 I think that this is the age of her target audience judging by her description of an irrational number.

  15. #15 rpenner
    October 19, 2009

    Because that same “subtle point” holds for the existence of any human thought, especially mathematical concepts.

    Just because they are encoded in the firing patterns of neurons, and in the patterns of language, doesn’t mean that they exist.

  16. #16 Mark W
    October 19, 2009

    I kept thinking about this quote: “God made the integers; all else is the work of man” by Leopold Kronecker. If you take out the last paragraph concerning Pi, then it reads more plausible. No?

  17. #17 John Armstrong
    October 19, 2009

    Interesting. You espouse a strong antiplatonist stance. But even Kronecker said that God created the integers.

  18. #18 Robert
    October 19, 2009

    Mathematicians treat the natural numbers as if they exist independently from humans in the sense that the naturals have properties that humans can discover, but humans cannot define or change the properties.

    All other numbers and the operations performed on those numbers are defined by humans.

  19. #19 Brian
    October 19, 2009

    I think mathematicians often say that “God” created the numbers more as a shorthand way of saying that human being didn’t create them; that they have an existence independent of us. (Denyse probably intended the stronger statement, though.)

    Mark, I think that her sentence makes marginally better sense if you just insert a comma in the middle: “All numbers are – in my view – created by God to march in a strict series[,] or else a discoverable* series.” I read the first half as referring to the integers, and then the second half brings in the rationals and irrationals, such as pi.

  20. #20 Joshua Zellinsky
    October 19, 2009

    I’m not sure I agree with your take here. I agree that numbers aren’t the work of a deity that the descriptor given by O’Leary is once again wrong. But it isn’t an egregiously wrong position. Indeed, it isn’t clear to me what it would mean for a universe to exist with different numbers. But that’s connected to the issue that I can’t conceive of a universe where basic logic worked differently. But that’s not to say that it couldn’t exist in some sense. The general human inability to conceive of something is not a priori a reason to assert its non-existence.

  21. #21 Nemo
    October 19, 2009

    A lot of very serious mathematicians, past and present, would disagree with you strongly. (Among them Paul Erdös, if Wikipedia can be believed.)

    You seem to have stepped into one of the longest-running debates in the philosophy of mathematics without even realizing it (?).

  22. #22 Badger3k
    October 19, 2009

    I get that argument all the time, that numbers have some kind of existence by themselves. The people arguing can’t seem to grasp that the number 2 would not exist without people. There would be “2″ objects, but unless there was other intelligent life to put words to the concept of “x number of objects,” there wouldn’t be any number 2. The same with arguments for other concepts. I think it usually comes down to some Platonic ideal or else the idea of “well, God exists, therefore His mind has the number 2 in it, so it has an existence” (or something – the argument never made sense to me, so I’m probably not giving it properly).

  23. #23 Scott LaBounty
    October 19, 2009

    Nice article, small nit though. I’m guessing you didn’t mean “piano numbers” but “peano numbers”.

  24. #24 Nobody Important
    October 19, 2009

    Also, isn’t the woman’s name Denyse O’Leary?

  25. #25 Scott
    October 19, 2009

    While I have no problems with you making fun of Denise, Nemo is right. You’ve stepped into a serious debate and taken sides, and you’ve represented your side as obvious truth without even a mention of those who disagree with you (such as Godel, for example). Whether or not numbers exist is a philosophical question most mathematicians don’t care about.

  26. #26 Robert
    October 19, 2009

    The use of numbers for counting is a human abstraction. But the natural numbers had properties that existed before humans discovered their use for counting.

    For example:

    1 + 2 = 2 + 1

    is a property of positive integers that humans discovered, humans did not invent it.

    That’s what they mean by “independent existence.”

    On the other hand, negative numbers did not have properties before humans defined them. There is nothing in nature to tell us:

    -1 X -1 = +1

    That is a rule of a human-defined operation that does not exist independent of humans.

  27. #27 Jason Orendorff
    October 19, 2009

    Of course it is crazy to say that π exists as a physical object in nature, any more than buildings or bees or blogs. None of these things are terms in the fundamental laws of physics. (Well, except π.)

  28. #28 zingle
    October 19, 2009

    Yeah man, but the equivalence class of things that can be put in one-to-one correspondence with the set {1,2} always existed before any human beings or other intelligent animals came along and thought about it or made such correspondences themselves.

  29. #29 Michael L Perry
    October 19, 2009

    I’ve enjoyed your blog for quite some time, but the past few posts have been nothing more than bile. I completely agree with your position, but I miss the “Good Math” half of the title. Best of luck.

  30. #30 BlueRaja
    October 19, 2009

    Are you ever going to write again about things we actually care about, or are you going to rename the site to “Good Religion, Bad Religion?”

  31. #31 Gray Gaffer
    October 19, 2009

    Um, @26, I always thought that 1 + 2 = 2 + 1 because the operation ‘+’ over the set of ‘integers’ is defined that way. By humans. It’s an axiom, not a discovery. And that -1 x -1 = +1 is also axiomatic. As in: define binary operator ‘-’ by the set of relations {.x.=., .x-=-, -x.=-, -x-=+} where ‘.’ is the absence of – (sometimes for convenience we use ‘+’ instead of ‘.’, but with the same absence meaning).

    Associating the set of integers ordered by the ‘<‘ operator with some arbitrary set of real world objects is also a human invention. A very old one too, since it made the difference between being rich or poor back in the very old days. Though the ‘<‘ way of describing those sets is relatively new. Again, though, naming the members of such a set required the human invention of names for human abstractions of thingness as distinct from specific bricks. ‘two-ness’ does not exist outside of human invention, and is never a property of real objects, only of human to human communications about real objects.

    In fact, I might be able to make a case that all mathematics is about communication, be it between humans or between myself and a future version of myself. It is about describing a certain class of concepts, and what is description if not communication. And communication of abstractions between humans is obviously (I hope obviously) a human invention.

  32. #32 Anonymous
    October 19, 2009

    @31

    I’m trying to follow Courant and Robbins “What is Mathematics?”

    The lowest level of abstraction is to start with natural numbers representing concrete objects and combine them by addition and multiplication the way you would take objects in boxes and combine them physically. That’s the level where you make discoveries such as the commutative laws.

    My understanding is that this level represents the first part of the famous Kronecker quote. When you go beyond natural numbers, and the addition and multiplication of them as if they were concrete objects, is when you get to Kronecker’s “everything else.”

  33. #33 eddie
    October 20, 2009

    I am corrently reading Lee Smolin – The Life of the Cosmos in which he contends that even the most basic logic is contingent on the universe having enough stucture in it to make a distinction between thing and not thing, let alone have entities capable of making such distinctions.

  34. #34 eddie
    October 20, 2009

    At first that seems to lead to a circular argument that (simplified) complex structure needs logic needs complex structure. The conclusion is that the universe *doesn’t* need logic, maths, physics to work. It just works, and logic is totally man made. I think Erdös, Kroenecker et al lose this round.

  35. #35 MPL
    October 20, 2009

    @ 18,

    Mathematicians treat the natural numbers as if they exist independently from humans in the sense that the naturals have properties that humans can discover, but humans cannot define or change the properties.

    All other numbers and the operations performed on those numbers are defined by humans.

    No, really, we* don’t. Mostly, we don’t think about it too much at all, but if we do, the natural numbers are just a particular algebraic structure. You can assert them as primary, or you can choose different foundations (sets, functions, successor operations, categories, etc). Really, whatever comes in handy at the time.

    Clearly, the counting numbers have a special place in all our hearts, but a lot of that is just that counting numbers are the ones which we learn first as children, are embedded in our everyday language, and have especially simple physical analogs.

    Put another way: we might think natural numbers are special, but math doesn’t**.

    * full disclosure: not yet a licensed mathematician, in training.
    ** illustrative metaphor: not approved for literal use.

    @29,

    What’s your problem with Mark’s enchiladas?

  36. #36 Robert
    October 20, 2009

    “But whether there is a God or not has nothing to do with whether A∧¬A == false.”

    To be honest, the only way I would be able to imagine an all-powerful god is if he were somehow able to set A∧¬A == true. If he were somehow more powerful than the laws of logic. Sometimes I think that’s what most religious people think too (if they thought hard enough about it) considering the huge amount of contradictions they believe in. God would be truly all powerful if he were able to bend the rules of logic in such a way that numbers, such as we think of them, come out to be a logical result from the axioms we chose.

    Of course, if one believes god is more powerful than logic, no amount of reasoning is going to have any effect.

  37. #37 Jan
    October 20, 2009

    A∧¬A == false

    Did God create logic? If you’re denying Platonism, why not go even further and look into paraconsistent logic?

    You might not like it, but I guess a large percentage of mathematicians has certain Platonist beliefs (though maybe secretly or unconsciously). This has probably partly to do with the fact that many of the really abstract objects that were invented (discovered?) throughout the centuries, first based on things happening in nature, but later purely based on other mathematics, turn out to be useful in describing nature.

  38. #38 Jonathan O'Connor
    October 20, 2009

    In Carl Sagan’s novel “Contact”, the extra-terrestial civilizations have a belief in a god that is able to define pi with a sequence of digits (in base 2, but I don’t think that is explicitly mentioned), millions of digits into the expansion, that form a rectangle (with primes as the lengths of its sides, of course) of mostly zeros, with a few 1 bits, which when looked at form a circle! The atheistic heroine of the book eventually finds this sequence, and that’s how the book ends.

    I like this idea, and I seem to remember, that if you look long enough at the binary/decimal expansion of pi, that eventually, you will see any given sequence of digits you desire. Is that true? Or is it an unanswerable question?

  39. #39 Christophe Thill
    October 20, 2009

    “Far worse, there is absolutely no way of claiming that π really exists. There are no perfect circles in the universe. And the only sense in which π can possibly exist in the real universe is as a measurement.”

    If I can be allowed a little nit-picking here:
    I disagree… not because there are no perfect circles, but because there are no perfect measurements. Measure the length of a circle (with radius = 1) with the best instruments, and all you’ll get is a number with many decimals, perhaps, but not an infinity of them. It will only be a rational approximation of π. I’d say that “the only sense in which π can possibly exist in the real universe” is as a limit: if you can improve indefinitely the quality of your measurements, you’ll get ever closer to it. You’re right, it’s a concept and nothing else.

    Some people really are clueless about maths. (And biology, too…)

  40. #40 wice
    October 20, 2009

    #38: damn, i just wanted to mention it (i mean ‘Contact’ by Sagan). btw, it ruined the book for me, and i cannot understand, what Sagan wanted with it. i’m pretty sure that pi is not something for which you can just decide a value, and then create a world, where pi has that specific value. pi “is the ratio of any circle’s circumference to its diameter in Euclidean space” (from the wiki), and it is so, no matter what kind of world you are living in.

    of course i’m open to any proof to the contrary, but i won’t hold my breath.

  41. #41 Bob O'H
    October 20, 2009

    Yes, as Scott asked, what’s with the Piano numbers? Are there only 88 of them, counted in base 13? :-)

    More on topic, Denyse is seriously confused about a lot of things. This just adds to the list.

  42. #42 Mark C. Chu-Carroll
    October 20, 2009

    @39:

    Where are you going to find a perfect circle to measure?

    In our universe, there’s no such thing as a perfect circle. Even if you had infinitely precise measuring tools, there’s nothing that you could measure that would produce exactly π.
    It exists only as a mathematical formalism. It’s an valuable mathematical formalism which describes lots of interesting and important things. But nowhere in the universe can you find anything whose value is π.

  43. #43 Mark C. Chu-Carroll
    October 20, 2009

    @38:

    I really, really hated the end of Contact.

    There are tons of constants in the universe that are, in some sense, variable. They’re properties of the shape or structure of the universe. If Sagan wanted to have a message from God encoded in the universe, it would be easy enough to do it – the ratio of masses of the fundamental particles, or the value of the cosmological constant for two examples. But instead, he chose a value which can’t be changed or shaped.

    Hell, he could even have used the value of a measured π – if space isn’t completely flat (and it isn’t), then the measured value of π would vary slightly from the value of the theoretical π.

    But being a physicist, he knew that even with the best imaginable instruments, measurements have a small number of significant digits. You just can’t get the thousands and thousands of digits that he needed for his story.

    So rather that doing bad physics, he did bad math. It pissed me off.

  44. #44 Jud
    October 20, 2009

    There is nothing in nature to tell us:

    -1 X -1 = +1

    Really? If I reverse the operation of taking one object away (do “negative one,” “negative one” times), isn’t that adding one object (“positive one”)?

    Mark: Denise wouldn’t buy a house where a satanist had performed his phony rituals without getting it purified by a priest with holy water, and…she wouldn’t see anything remotely silly about it. She sees her superstitions as legitimate, but others as mockable.

    So how do you feel about mezuzot? Legitimate or mockable? Legitimate for you, OK to mock for others?

  45. #45 Jud
    October 20, 2009

    I wrote: So how do you feel about mezuzot? Legitimate or mockable? Legitimate for you, OK to mock for others?

    That reads on the page as more hostile and challenging than I intended it to be. I am actually sincerely curious as to the answer, if you’d care to share it.

  46. #46 Matt Springer
    October 20, 2009

    “I like this idea, and I seem to remember, that if you look long enough at the binary/decimal expansion of pi, that eventually, you will see any given sequence of digits you desire. Is that true? Or is it an unanswerable question?”

    It’s an open question. Most mathematicians think it’s true for a number of good reasons, but so far no dice in terms of proof. The formal question is whether pi is a normal number.

  47. #47 Mark C. Chu-Carroll
    October 20, 2009

    @45:

    How do I feel about mezuzot? That depends on what the person who hangs them thinks.

    Some people hang them because they believe that they’re lucky. I think that’s stupid and mockable. Thinking that hanging a little scroll of paper in your doorway is going to make good things happen to you, and neglecting to hang one is going to make bad things happen? That’s every bit as silly as insisting on buying a house that faces 5 degrees west of north, or refusing to buy a house whose address is #13.

    Some people hang mezuzot because it’s a reminder. Jews believe that there are rules that you should follow – not because following the rules is going to make anything good happen, but because following the rules is the right thing to do. Mezuzot are a visible reminder of those rules – and you see that reminder every time you enter or leave a home with the mezuzah on the doorpost.

    I’m in the second group. I see the mezuzah as something to make me think about what’s right, and what’s wrong, and what I should be doing.

  48. #48 Nelson
    October 20, 2009

    @38

    I think you can take some convergent infinite series , you can reorder the terms of that infinite series to converge to another set value. So if I have an infinite series which converges to Pi, I could take the numbers which define the series and reorder them to instead converge to e (or 1 or 2 or any number).

    I think the criteria for that reordering is if it is an infinite convergent alternating series… though that might be too strict a condition. I think the question is whether it is still the same series after you reorder the terms which depends on whether or not its uniformly convergent. I could be wrong, I’m going off of memory.

    Sorry if that’s off topic, that’s what I thought of when I saw your post.

  49. #49 Nate
    October 20, 2009

    Some of these comments are going off the deep end into metaphysics and similar wishy-washiness. An abstract formalism of numbers is just that: abstract. Abstract concepts don’t have a physical, tangible existence. However, there are some physical, tangible objects that exist in the universe whose behavior can be modeled – by humans – with numbers. We found that our beefed up system of logic can be used to do nifty things, like keep track of how many cattle we’re accumulating or to predict how cold it will feel tomorrow. But cattle don’t reproduce dependent on a definition of natural numbers, and the temperature is a function of particle motion, not the real number line.

    Mark’s “no such thing as exactly pi” doesn’t even enter the question; many numbers do appear in our models exactly, be they natural numbers or not. We can describe abstractly various universes with various rules and then use numbers to build a model with which to predict grander things about them. The universe is “built” around the rules, not the model.

  50. #50 Cyan
    October 20, 2009

    @48

    The criteria for a conditionally convergent series are that the series must have an infinite number of both positive and negative terms, and the series formed by taking the absolute values of the terms in the original series must diverge. (Or equivalently, the positive terms taken alone diverge to positive infinity and the negative terms taken alone diverge to negative infinity.)

    Such a series can be re-ordered to converge to any (without loss of generality, positive) target value but alternately summing positive terms until the target is overshot, and then summing negative terms until the target is undershot.

  51. #51 Anonymous
    October 20, 2009

    I wish more believers were as honest as you are… I wish more unbelievers were as honest as you are… :)

    You get 100 cool pts!

  52. #52 Nelson
    October 20, 2009

    @50

    Yes, I remembered that the process involved over and under shooting which is why I said alternating series, but that’s definitely not general enough. As you said, infinite number of positive and negative terms while the absolute value of the series converges. Thanks for reminding me.

  53. #53 Nelson
    October 20, 2009

    Erp, strike aboslute value converges – replace with diverges. I’ll just be quiet now. :)

  54. #54 Marko
    October 20, 2009

    The ending of Sagan’s book »Contact« bugs me too.

    »The Argus computer had gone deeper into π, deeper than anyone on Earth, …« but the »anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and ones«. (I have the paperback version in front of me.)

    Further: »The program reassembled the digits into a square raster, an equal number across and down. The first line was an uninterrupted file of zeros, left to right. After a few more lines, an unmistakable arc had formed, composed of ones« and so on.

    Having read it again after 10 years, I’ll try to code a circle search for π — well, if you can call those jagged paths of ones a circle. In the simplest form, the bit sequence

    0 0 0 0 0
    0 0 1 0 0
    0 1 0 1 0
    0 0 1 0 0
    0 0 0 0 0

    would satisfy Sagan’s description for a representation of a circle with radius 1. Occurances with larger radii would no doubt be less frequent, but still somwhere »inside« π.

    Unclear to me: Are the »circles« supposed to be hollow or filled with ones? And: is garbage between each scanline allowed? This search resembles more and more the infamous Bible Code search, alas with a »scripture« text of infinite length and variation.

  55. #55 Marko
    October 20, 2009

    Just wanted to add the obvious: while finding patterns in π like those Sagan described would be be cool, and all the awesomer, the larger the radii are; it wouldn’t mean that some intelligence put it there.

    It’s just the law of large numbers combined with the fact that π is an algebraically independent transcendental number (if not normal, which still is an open question, yet probable and plausible, as I understand).

  56. #56 Maria Droujkova
    October 20, 2009

    I really enjoyed the parts deconstructing the “marching in series” statement. I think it would be even stronger with less meta-discussion.

    I also read that sentence in a way that makes a perfect sense to me, when I replace MY “all numbers” by Denise’s “all numbers” – or maybe by her demographics, tribe, what have you. No, they don’t deal with all the wonderful weirdness of numbers, ever. All that subtlety and complexity is completely invisible, practically non-existent to them. These people deal with natural, rational, and a few select irrational numbers like Pi, and that’s probably it. It reminds me of the ethnographic studies of some tribes that count, “One, two, many.”

    To me, there is an issue of whole cultures or demographics having their own alternative or limited math worlds. Do we mock them? Do we mandate that they learn all the “number weirdness” we know and love? Do we say it’s “cultural differences” and celebrate the diversity?

    This is another philosophical question to add to the growing list. What should we do about groups of people effectively living in different math worlds – say, the world where most numbers they actually meet are rational?

  57. #57 Charles Tysoe
    October 20, 2009

    This is a really entertaining and informative discussion! I think some have read too much into Ms. O’Leary’s comments on number, surely not made to posit a formal mathematical postulate! It seems to me rather to be saying that order and number in the universe are mutually dependant, that this relationship is discoverable, and it is far from unreasonable to infer the existence of a transcendant, powerful, thinking,
    planning God. In fact mathematicians have been known to give praise to the Creator for the number embedded in nature. Such statements in themselves did not render their theories true or false, testing is the only real way. Conversely, a series of errors on the path to a true solution would not render that mathematician’s belief in God false or insane.

    Moreover, Mr. Chu-Carroll, where is your proof that God didn’t “make” numbers? Are you entitled to it by virtue of your mathematics prowess? Is this a problem so self evident it requires no proof; like 1+1.!

    You misspelled your adversary’s name, so it is plausible other simple realities elude you! ;)

  58. #58 Charles Tysoe
    October 20, 2009

    “Numbers were not created by a supernatural being. No deity, no matter how powerful, could have created a universe where numbers didn’t exist, or didn’t work.”

    Also, the meaning of this statement eludes me. Is the second supposed to follow from the first?

    I’ll try: “The most powerful conceivable deity could not create a universe which did not contain and function according to numbers. Therefore conclude that our universe, which contains numbers which work, was not made by any conceivable deity”

  59. #59 Jud
    October 21, 2009

    Re #s 57 & 58 – I haven’t had any math practice since high school calculus nearly 40 years ago, but if I can take a crack at what I understand Mark to be saying (surely he and/or others will correct as necessary), it is that there are no such things as numbers in the Universe. Numbers are abstract concepts that can be used to describe real physical phenomena, but are not themselves to be confused with the phenomena they can be used to describe.

    In fact, not all numbers we know of have any sort of correspondence to real physical phenomena; and beyond that, there are infinitely many numbers we haven’t even found, let alone used to describe anything, in any category of what we define as “numbers.” So it isn’t a question of whether a God would have the power to create something, it’s that there’s no “something” to create.

  60. #60 Jud
    October 21, 2009

    Much better than my lame attempt above, read the Wikipedia articles on real and complex numbers, particularly with reference to Ms. O’Leary’s description of numbers as being in a series, and also regarding correspondence (or lack of same) with any physical reality.

  61. #61 bob koepp
    October 21, 2009

    A number of people have said, in effect, “Numbers are abstracta, not existents.” Is there a false dilemma lurking here?

  62. #62 Orin T.
    October 21, 2009

    “God created the integers” is the first part of a quote. “the rest is the work of mankind” finishes it.

    Many have woundered about the unreasonable effectiveness of mathematics. Penrose found his solution in the Platonic forms.

    Also some, such as Hardy, were proud of the fact that their math had no practical applications. After he died practical applications were found.

    “Is God a Mathematician” is a good book on the subject.

  63. #63 Escuerd
    October 22, 2009

    rmp @ 1:

    That reminds me of a time when a physics professor was explaining that the operators for quantum mechanical observables had to have real eigenvalues.

    “You won’t ever measure a complex position or energy. They aren’t real.”

    The class burst into laughter, as he realized what he had said.

  64. #64 Escuerd
    October 22, 2009

    And yes, I know that was only tangentially related (maybe in that the prof came close to making the incorrect implication that real numbers are more real than imaginary ones), but I thought it was funny enough to share anyway.

  65. #65 M Dj
    October 22, 2009

    While I agree with your post in general, there are a couple of clear mistakes you make, so it’s pretty rough to mock people the way you do without having all the facts yourself. First of all, you say:

    “Numbers were not created by a supernatural being. No deity, no matter how powerful, could have created a universe where numbers didn’t exist, or didn’t work. ”

    Let me first say that I don’t believe in God. But the problem with your sentence is you don’t comment on whether God exits or not, but you say that if God existed, and no matter how powerful he was, he couldn’t have created a universe where numbers didn’t exist. This is not true. A simple example would be just a completely empty universe, with nothing inside. Surely an all-powerful deity could create that! I don’t see how numbers would exist there. A more non-trivial example would be a universe where everything was just an amorphous blob, and even if there were sentient beings, you could imagine beings whose consciousnesses are mixed with each other and intermingled, so that there is no way of talking about separate entities. If there is no way of separating things from each other, or talking about one thing vs another, then how could numbers exist in that world or work? And if I can think in 5 minutes of universes with no numbers, why couldn’t an all powerful deity create them? The only difference is that I don’t believe that deities exist.

    Now the second mistake is you say that there are no perfect circles in the universe. This is also not true. The horizon of a black hole can be in a state that is a perfect circle. Another example would be a state of a closed string – these can also be perfect circles.

  66. #66 Marko
    October 23, 2009

    M Dj: Even a completely empty universe would constitute 1 (one) existence, one spacetime and two types of symmetry: isotropy (directional symmetry) and homogeny (translational symmetry). Either way, there would be a countable amount of minima and maxima of something, e.g. of zero-point energy or space-time curvature, due to virtual particles.

    The same is true for the second model you’re proposing, the amorphous blob universe (I like that concept). What’s more, the blob has to be made out of something. Æther? (-;

  67. #67 gbk
    October 23, 2009

    Yeah, superstition brings bad luck.

  68. #68 M Dj
    October 24, 2009

    @ Marko:
    The empty Universe can be compact and doesn’t have to contain any symmetry. There would be 1 spacetime, but you see you’re just applying the concepts of our Universe to the description of another (hypothetical) Universe. But in that other Universe there is no mechanism by which the concept of numbers emerges, since there are no quantities to be counted or compared, or anyone to compare them.

  69. #69 Jonathan
    October 24, 2009

    I’ve got to pull you up on the comment:

    “Numbers were not created by a supernatural being. No deity, no matter how powerful, could have created a universe where numbers didn’t exist, or didn’t work.”

    The second sentence seems to have no grounding as far as I can see. True, we can’t imagine a universe in which if you were placed there as a probe numbers would not exist. I’d simply claim that because we can’t conceive of a universe where numbers don’t exist, we also can’t conceive of a deity creating such a universe. This seems to be a fault of our own imagination and not of the power of an all-powerful creator.

    In fact the first of the above sentences seems to be based on a bias that in a universe which was created by a deity there are external things of which it had no control. This seems simply to be a particular point of view. If you’re going to dismiss someone’s opinions you need to look at your own biases for belief.

  70. #70 p
    October 25, 2009

    It can’t account for logical wierdness like Gödel’s incompleteness theorem, which relies on the logical structure of numbers.

    fwiw, Gödel (and arguably Cantor) believed firmly that the human mind can, in principle, resolve any rational question that it can pose. In his view, his incompleteness results only demonstrated the poverty of the formalizations we have, and there should be no reason we could not develop stronger methods which could solve all problems we could pose. (“Stronger” in a more fundamental sense than e.g. strengthening the Peano axioms with ad hoc axioms which solve whatever particular problem we might consider.) This is fairly surprising from the point of view of the popular conception of Gödel, but he devoted much of his life to such considerations. His method of attack was apparently based in Husserl’s phenomenology.

    There is a good paper on this very topic I happen to be (slowly) reading right now in the Bulletin of Symbolic Logic, I believe it’s posted online. The title is “Gödel’s Program revisited”.

  71. #71 p
    October 25, 2009

    I’d simply claim that because we can’t conceive of a universe where numbers don’t exist, we also can’t conceive of a deity creating such a universe.

    Isn’t inconceivability supposed to be an attribute of God? (c.f. Job)

  72. #72 Tomato Addict
    October 25, 2009

    This would be a great post even with no mention of O’Leary silliness, but poor Denise gets PWND anyway, making this a double treat!

  73. #73 Vorlath
    October 25, 2009

    I can’t believe there are still people who believe in Cantor’s theory. It’s complete silliness and makes the people who believe in superstition look like the smart ones by comparison. Cantor was well known to treat infinity as a finite number and that’s all he’s doing with his theory. It doesn’t mean anything.

  74. #74 S.C. Kavassalis
    October 25, 2009

    Yes, her name is Denyse O’Leary – her blog post that this was quoted from is here: http://post-darwinist.blogspot.com/2009/10/mathematics-gap-tooth-creationist-moron.html . I’m also pretty sure that, while the Discovery Institute does reference her “work”, she’s not officially associated with them.

    Now, I do agree that her statement is entirely ignorant about mathematics, but as creationists go, she at least tries to exist on the reasonable side of things. I have to respect her a little bit, for, in other works, trying to come to terms with the multiverse theory in terms of religious belief (which I haven’t really seen too many try to do). As an ardent atheist, I think she’s wrong and very ignorant of fundamental physics, but I at least respect her for being a tiny bit more consistent than most.

    O’Leary’s viewpoint (despite her not being familiar with the total ordering or well-ordering of the reals, which is really hard to fault a lay person for) is not all that dissimilar to that of a mathematical Platonist. Platonists believe that all mathematics, all numbers, truly exist in the universe, and we are only discovering them as we go, not inventing. As a formalist, I think this is silly, but it’s still a widely held belief, by members of the mathematical community even. Mathematical constructivists even exist, as practising mathematicians, where the only numbers (or functions, or well-orderings, etc.) that exist are those that we can describe. The philosophical discussion to completely dismiss mathematical Platonism and Constructivism are more subtle than you might want to admit (and I say this as someone who disagrees with both of those points of view).

  75. #75 p
    October 25, 2009

    @73: Whether one “believes in” Cantor’s theory or not hasn’t much to do with the practice of mathematics. Fascinating work continues to build on it and will probably continue indefinitely. There is a book summarizing work from this century called “The Higher Infinite.” You might find it interesting whether you regard it as “real” or mathematical “science fiction”, either way, it’s still fun stuff.

    As far as whether it “means anything” outside of itself, that is a subtle issue and a hard case to make either way, even for the integers, as the debate on this thread demonstrates.

  76. #76 TruePath
    October 30, 2009

    Umm, assuming choice every set can be well ordered. So you can “list out” the reals in a certain sense.

  77. #77 mark
    November 4, 2009

    1) God created the universe.
    2) The universe contains the concept “numbers”.
    Therefore … God created “number”.

  78. #78 grenuck
    November 12, 2009

    Such a judgmental lot. It kills me how this group seems to pride itself on how “smart” it is simply because you are able to blather on about something someone else (probably a professor or text) taught you about numbers. We are all equally stupid and no matter what you are an expert on you are being called stupid in a multitude of parallel universes. For instance, here you mock people for their ‘infantile’ understanding of numbers and number theory while in a parallel blog people are mocking your use of the English language or your ignorance of philosophy or religion. Or they are making fun of how much energy of your lives you have wasted on a topic that doesn’t change anything for 99.99% of the world.
    By the way God must have made numbers because had anyone checked the bible they would know that Numbers is between Exodus and Deuteronomy. (that’s a joke — please go on wasting your life with blogs about how stupid people seemingly don’t know the same things you do)

  79. #79 Clif Davis
    December 3, 2009

    An ignorance about nuances of the English language and a widespread ignorance of basic underlying issues of philosophy or religion are probably not reasons for alarm. Basic innumeracy is, particularly on the part of the decision makers of our society We live in a universe that is increasingly dominated by mathematical facts of life.

    The title of the Blog should have given you a clue that involved a critique of the use of mathematics. The value of blogs like this is that they provide some education along with their critiques. And if there is some one-upmanship in the comments, that has to do with being human. Consider disagreements a chance to do at least a Google search to further your understanding, or go read blogs that irritate you less.

  80. #80 T. Cook
    April 21, 2010

    “Anyway, I didn’t get a chance to say this on the program so I will now … All numbers are – in my view – created by God … I tend to flub the gap-tooth creationist moron role…” -Denise O’Leary.

    Denise didn’t stand up for her beliefs in this interview. She had an idea of what they expected of her and withheld those views. I wonder why she would do such a thing–given how much more acceptable it is to be theistic than atheistic in America. I can only conclude that she knows how silly of a notion it is so she saved it for preaching to her choir. At least she didn’t subject her metro area to that unfounded propaganda.