Annals of Bonology, Vol. I, No. 1, 1-5, 2016
Bones: How Many Are There?
An Overview of the History and Practice of Bonology
Sean McAfee1 , Illustrations by Jennifer Kenkel2
Abstract
Bones. We are all pretty sure what they are; for example, we have arm bones, leg bones, and even a head bone
(or ’skull’). But exactly how many bones do we have? Clearly there are the five just listed, but once we begin to
consider finger or toe bones we quickly lose count. What can be done? While this problem may seem intractable
at first, there is a small but growing community of scientists who are dedicated to developing methods to deal
with the numerous obstacles inherent in bone counting. Leading the charge is former orthopedic surgeon Steven
Haskler; in the three months since he abandoned his practice to establish this new branch of science (aptly
named ’bonology’), he has made great strides in both technique and theory. This article attempts to give the
layperson a gentle introduction to this fascinating but frequently overwhelming field of study.
Keywords
Bones — Many — How
1 Unemployed,
2 Between
Salt Lake City, United States
jobs, Salt Lake City, United States
Contents
Introduction
1
1
History
1
2
Techniques
2
3
In Search of the Bone Constant
3
4
Ethical Issues and Controversy
4
5
Results
5
6
The Future of Bonology
5
Appendix: Proof of the Fundamental Theorem
6
Introduction
”I truly believe there is no bone which cannot be counted.”
These are the optimistic words of Dr. Steven Haskler, speaking
from a makeshift podium of orange crates at the first annual
Bonology Conference, held at the Institute for Bonological
Advancement in Baltimore, Maryland. The crowd is sparse;
there are roughly two dozen in attendance. Some are listening
intently, others seem distracted or twitchy. Still others are
visibly drunk, teetering precariously on their folding chairs
or overturned buckets. This audience is a reflection of the
diversity of both people and ideas in the emerging and exciting
field of bonology.
What is bonology? Roughly speaking, it is the study of
quantity; namely, the quantity of bones in a given human
being’s body. While the definition of bonology can be stated
in a few words, the difficulties involved in its study begin
almost immediately. First and foremost, there is vigorous
debate as to what exactly constitutes a bone. For example,
is the nose a bone? ”Of course not,” says Dr. Paul Murphy,
director of oncology at Johns Hopkins Medical, ”it’s just a
piece of cartilege. Bones are composed of calcium phosphate
and calcium carbonate. Is that what you came here to ask
me?” Some, like Haskler, would politely disagree: ”If the
nose is not a bone, then why is it kind of hard, like a bone?
And why do we call it a ’broken nose’ if there is no bone to
break?”
There is also healthy debate between bonologists and nonbonologists about research methods. Most notably, many
outsiders claim bonologists’ practice of stealing bodies from
graves under cover of night in order to count their bones is
disrespectful to the dead; bonologists assert this is the cost
of scientific progress and that we should not let superstitions
inhibit our research.
Even amongst fellow bonologists, there is some disagreement as to whether each person’s body contains the same
number of bones. Those on one side of the argument point
out that common sense would indicate a person weighing 200
pounds should have roughly twice as many bones as a person
of 100 pounds. Others postulate the existence of a ”bone
constant”: a fixed number that would precisely describe the
number of bones in any person’s body.
Before addressing these and other thought provoking aspects of bonology in detail, it is best to understand the emergence of bonology as a hard science. Its origins lie in Baltimore, Maryland with the work of Dr. Haskler in the early
winter of 2015.
1. History
Although the question of how many bones we have has captured the imagination of both scientists and philosophers alike
Bones: How Many Are There?
An Overview of the History and Practice of Bonology — 2/7
Figure 1. The Institute for Bonological Advancement
for millenia, it was not until November of 2015 that the first
serious inquiries were made by Dr. Steven Haskler. ”We had
just had our first big snowfall of the year,” recalls Haskler. ”I
got to work early, and I guess they hadn’t had time to throw
any salt down yet because on my way to the hospital from the
parking lot I slipped on a patch of ice and just...” he pauses
with a chuckle and hits the table with his palm for emphasis
”knocked myself out cold! Well, I’m not sure how long I
was lying there on the sidewalk unconscious, but I remember
waking up with a weird sense of giddiness. Also nausea, and
blurred vision. It was like I could sense that this day would
be special in some way.
”I made my way to my office and began to look over
patient files, but for some reason I just couldn’t concentrate.
My mind kept wandering, and as my eyes drifted around the
room I found myself staring at a model of a human skull on
my bookshelf propping up a copy of Gray’s Anatomy. ’That’s
one bone’, I remember thinking, ’but how many more?’
”I wandered into the hallway, ears ringing with excitement.
After almost bumping into a coworker, I asked, ’Hey, have you
ever thought about the possiblity of counting all of the bones
in the body?’ She just laughed and and kept walking down the
hall. After getting similar responses from other colleagues, I
realized that I was onto something. The people in my field
had clearly convinced themselves that bone counting was a
fool’s errand, and I was determined to prove them wrong.”
Haskler left work that day and hasn’t looked back since.
Since then, he has spent countless hours laying the groundwork for this fledgling science. His center of operations is the
Institute for Bonological Advancement (see Figure 1). Nestled in an abandoned warehouse near the intersection of I-95
and the Baltimore Beltway, it provides ample research space
for Haskler to develop theory and refine his methods. When
not at the Institute, Steven works tirelessly to rally others to
the cause of bonology, wandering through parks and riding the
public transportation of Baltimore in search of like-minded
individuals to aid him in his investigation.
2. Techniques
Though there are many facets to the study of bonology, the
central problem can be summarized as follows: given a pile of
bones, how can we estimate (or, ideally, calculate) the number
of bones in the pile? Most of Dr. Haskler’s work has relied
on utilizing what he calls ”The Fundamental Theorem” (see
the Appendix for a proof):
β = α +ω +ε
(1)
Here, β represents the total number of bones in a given
pile, α and ω indicate the number of bones already counted
and those that remain to be counted, respectively, and ε is
an error term which accounts for such things as miscounted
bones or bones accidentally taken from another pile. In terms
of this equation, the work of the bonologist is twofold: he
must calculate α and ω through various counting methods
discussed below while at the same time minimizing ε.
Early attempts at calculating β involved the bonologist
pulling a bone from the pile, saying ”one”, putting the bone
Bones: How Many Are There?
An Overview of the History and Practice of Bonology — 3/7
back in the pile, pulling a new bone from the pile, saying
”two”, and continuing in this fashion with the intention of
eventually exhausting all uncounted bones and arriving at a
total. While seemingly foolproof at first glance, this prototypical method actually contained two subtle flaws which needed
to be addressed.
The first methodological hurdle stemmed from the fact
that the act of counting out loud was prone to the bonologist
losing track of which number they had most recently spoken.
A bonologist’s laboratory can be a hotbed of activity, with
many distractions; the roar of passing freight trains or the
occasional belligerent drifter wandering into the lab can make
it nearly impossible to maintain the concentration necessary
to keep an accurate bone count in one’s mind. This prompted
the invention of a technique called ”notching” (see Figure 2),
wherein a lab assistant (or ”bonehand”) will make a vertical
mark on a chalkboard or other surface as each bone is held up
for view by the bonologist. The recorded vertical marks (or
”bonograph”) can then be tallied to arrive at the desired β .
Figure 3. a counted bone, as compared with a bone which
has yet to be counted
Figure 4. illustration of the piling method
Figure 2. the notching process
Even after the introduction of the notching process, there
was shown to be a second, more fundamental obstacle to
reaching an accurate count using the above method: that
of distinguishing an uncounted bone from a counted one.
Once a bone was counted and placed back in the pile, it
became indistinguishable from the other bones left to count
(see Figure 3). As a result, a bonologist would often spend
hours on a count, never reaching the end of a given pile; bone
tallies in the thousands would often be reached before the
bonologist threw up their hands in frustration.
The solution was the introduction of ”piling” (see Figure
4), wherein counted bones would be placed in a pile separate
from the original pile of uncounted bones. In this way, the
bonologist could avoid the double counting which plagued
the bone tallying process.
The emergence of this refined counting method represented a sea change in the field of bonology. The basic ques-
tion still remained, however: how many bones are there?
3. In Search of the Bone Constant
As stated earlier, the central object of study in bonology is a
pile of human bones; to fully understand this pile, a bonologist
attempts to count the number of bones in it. The technical
advances outlined in the previous section all but eliminate
errors in counting (assuming the pile in question is a complete
specimen-an issue we will address later). We may therefore
consider a modified Fundamental Theorem: the lack of an
error term allows us to rewrite
β = α +ω +ε
as
β = α + ω.
Bones: How Many Are There?
An Overview of the History and Practice of Bonology — 4/7
This revised equation illustrates the elegant symmetry underlying the field of bonology: given the number of counted and
uncounted bones in a pile, we simply add the uncounted bones
to the counted ones to arrive at our total β . In particular, note
that as ω approaches 0, we see that β approaches α; i.e. once
we have completely counted the bones we arrive at a fixed
number denoting the number of bones in the pile.
This might seem to be the end of the story; indeed, some
(including Dr. Haskler) view the goal of bonology as a simply
applying the Fundamental Theorem to various piles of bones,
logging the results, and accumulating a large body of resulting
counts for use by future scientists. Other bonologists are more
ambitious in their research, going so far as to conjecture that β
is a fixed number, independent of the human being associated
with the pile of bones! Scientists of this opinion denote this
fixed number by B, and refer to it as ”the bone constant”.
This conjecture has sharply divided the field; Haskler in
particular is very vocal about his doubts: ”It is ridiculous
to assume such a constant could exist, given the complexity
of the human body. While human beings in general are of
the same genetic makeup, there is a large variation of traits
between individuals: eye color, skin color, height, weight,
blood type, predisposition to various diseases. It is naive to
assume that bone number would somehow be constant among
all people. In fact, I can give an immediate counterexample
to those proposing a bone constant: imagine identical twins,
and suppose one of them were to have an accident wherein
their arm was broken. At that point, the twin with the broken
arm would have one more bone than the other. So, if the
conjecture is false even for the extreme case of genetically
identical humans, how can it possibly be true for the entire
species?”
Others disagree: ”206. There are 206 bones in an adult
human body,” says Dr. Paul Murphy. ”How do I know?...I
mean, we’ve known this for centuries, basically since the
science of anatomy was created in the middle ages. There are
hundreds of books detailing each bone and its corresponding
function in the body. Seriously, I have a very busy job to do
here, and I’m not sure what the point of this interview is...wait,
did you say Haskler? You’ve been talking to Steve? Oh my
God, is he all right?! He disappeared without a word; we’ve
looked everywhere, filed a police report...you need to tell me
where he is!”
Clearly there are strong emotions on both sides of the
debate. Dr. Haskler is undaunted by this difference of opinion,
however: ”At the end of the day, we are all here for the same
reason: our love of counting bones.”
4. Ethical Issues and Controversy
While the underlying theory of bonology can be easily explained to the layperson, the logistics of being a practicing
bonologist may present some unexpected obstacles. In particular, in order to engage in research, a bonologist must have
a steady supply of piles of bones to count. The lack of acceptance of bonology as a discipline by the greater scientific
community has resulted in a lack of funding for countable
piles of bones. As a consequence, bonologists have been
forced to resort to sometimes desperate measures to acquire
research specimens.
Arguably, the most controversial of these measures has
been the practice of exhuming bodies from cemeteries for
use in the lab. Decried as ”grave robbing” by its detractors,
Haskler has a more nuanced view of the situation. ”Look, I
completely understand the perspective of those who are upset
by the collection process. It is in our nature to want to leave
bodies of the dead in place as a token of sentimentality or
respect; however, we cannot let the act of scientific discovery
be held back by these primitive impulses. I am sure that if the
families of the dead involved understood how essential the
bones of their loved ones were to our research, they would
be more supportive of our methods. It is really just a matter
of raising awareness.” Haskler added, wistfully, ”I imagine
this is how Galileo must have felt when he was criticized for
dropping bodies from the Tower of Pisa to test his theories of
gravitation1 ” (see Figure 5)
Figure 5. Galileo’s method
Even disregarding the public outrage, the practice of collection is in itself fraught with difficulty. Assume for the
moment that the set of bones to be exhumed is a complete set
(in practice, due to decomposition and other factors only a
fraction of the specimens collected have this property); there
is still the matter of physically removing the bones from the
ground. Although the average grave depth is only about three
feet (contrary to popular belief), this still leaves several cubic
yards of topsoil to remove manually. This can be a grueling
physical process which is further compounded by the fact that
one must also take into account the stealth required to accomplish the collection without alerting any graveyard security.
Furthermore, there is issue of moving the specimen from the
graveyard to the Institute for study.
1 ”We
Are Not Grave Robbers:In Defense of the Practice of Bonology”,
Steven Haskler, New England Journal of Medicine: rejected for publication
Bones: How Many Are There?
An Overview of the History and Practice of Bonology — 5/7
As a result of these challenges, bonologists have been
forced to enlist the help of others in the collection process.
This assistance may come in the form of participating in the
digging, carrying the coffin to a van kept running near the
graveyard, or keeping lookout for security or (potentially)
rival bonologists. Another key role of the assistant is in the act
of ”screening”, wherein the bonehand distracts employees of
the graveyard while the bonologist plies his trade (see Figure
6).
Figure 7. Ernie
The following summarizes the results obtained from Ernie
in three clinical trials:
”Ernie”
Trial A2
Trial B3
Trial C4
Figure 6. an example of screening
Unfortunately, those willing to assist the bonologist in
collecting samples tend to be unsavory or dangerous personalities. Many the bonologist has had their bonehand turn on
them after a succesful exhumation, robbing them of their valuables and those found in the opened grave. ”It is a sad fact that
our bonehands often do not share our sense of scientific wonder and are instead motivated by greed or violent tendencies,”
says Haskler. ”We do our best to weed out these applicants,
but ultimately it comes down to just hoping for the best and
having faith in your colleagues to do the decent thing and help
you dig up another pile of bones to count.”
5. Results
Although the science of bonology is less than two years old,
there have been several exciting (yet incomplete) results so far.
”To date, we have accumulated four specimens for study,” says
Haskler. ”Three of these were obtained from a local cemetery,
but upon closer inspection at the Institute the samples were
not the easily countable bone piles we expected...I’d rather
not get into specifics, but it was a bit disheartening, especially
after all the trouble we went through.
”The other specimen is a hanging skeleton which I was
able to acquire from a classroom at Johns Hopkins. It is nearly
complete, except for a leg that was lost while fleeing from
a security guard (see Figure 7 ). We affectionately call him
”Ernie” at the Institute, and he has proven to be our most
promising source of data.”
β
> 1323
> 37
> 117
α
1323
37
117
ω
β − 1323
β − 37
β − 117
”The third trial gives us a clear lower bound of 117 for
Ernie’s β ,” says Haskler proudly. ”Taking into account the
missing leg, which probably has at least 5 bones in it, we
can safely increase the lower bound to 122. As for an upper
bound, we can only make an estimate. At the time the police
kicked in the laboratory door, the pile of counted bones was
roughly equal to the pile of bones left to count; we can then
conservatively assume a β of no more than 300.
”This upper bound takes our study of bone numbering
from the realm of the potentially infinite and places it in
that of the finite. It is a remarkable accomplishment, even
if we have yet to reach a precise β for a given sample. Our
task is now to refine these results, raising the lower bound
while sumultaneously reducing the upper bound until the two
bounds meet and we arrive at an accurate total. Simple as
that.”
6. The Future of Bonology
So what does the future hold for bonology? Speaking from his
holding cell in the Baltimore City Detention Center, Haskler
2 This first trial predated the introduction of the piling method; the count
was abandoned after a total of 1323 counted bones.
3 This count was interrupted by a territorial dispute between Haskler and a
small group of hobos who claimed ownership of the Institute grounds. The
ensuing scuffle resulted in the piles of counted and uncounted bones being
scattered around the lab, ending the trial after a count of 37 had been reached.
4 The most promising trial yet: unfortunately, this too was interrupted by
the local police, who had received a tip from a disgruntled bonehand. At the
time of Dr. Haskler’s arrest, they had achieved a count of 117.
Bones: How Many Are There?
An Overview of the History and Practice of Bonology — 6/7
exudes optimism: ”This misunderstanding with the local police has been a blessing in disguise, really. It has allowed
me to take a step back from lab work and consider where
bonology fits into the bigger scientific picture.
”As far as I see it, the short term goals remain the same:
we need to accumulate as much data as possible and look for
general trends among various bone counts. Once that data has
been analyzed, we can move on to more ambitious projects.
For instance, until now we have only considered the β of humans; but what about animals? To start we have of course the
trivial case of snakes, which clearly have no bones. Slightly
more complicated is the one-boned turtle, which is essentially
a snake with a single shell-bone. Beyond that, though, there
is an entire world full of various boned creatures, with each
species presenting its own bone-numerical challenges.
”There is also the question of whether the techniques of
bone counting can be applied to non-bones as well. Perhaps
with some modification the techniques I have developed could
be applied to count such things as the number of cards in
a deck, the number of days in a year, or even, and this is
admittedly wishful thinking, the population of the planet!
Figure 9. quantum bonology
says Dr. Murphy, hastily grabbing his car keys from a drawer
in his desk. ”You can show yourself out.”
Appendix: Proof of the Fundamental
Theorem
Theorem (Haskler 2013): Given a pile of bones, let β denote
the number of bones in the pile, let α denote the number of
counted bones, let ω denote the number of uncounted bones,
and let ε = εα + εω , where εα , εω represent the innacuracy
(in number of bones) of α, ω, respectively. Then,
β = α + ω + ε.
Proof: For a given n ∈ N, we have the following inequalities:
β−
1
1
<α <α+ ,
n
n
1
1
ω− <ω <ω+ ,
n
n
α−
Figure 8. counting the unboned
”On the more theoretical side, we can ask ourselves the
following: what is actually occuring during the process of
counting? A bone transitions from an uncounted state to a
counted one, but is this transition discrete? Or is there some
sort of intermediate state, where a bone can exist as both
counted and uncounted at the same time? The answer to these
questions would necessitate the creation of a sort of ”quantum
bonology”, wherein the incremental act of counting would be
compared against the continuous flow of time.
”In short, I see a bright future for bonology,” says Haskler,
speaking above a guard knocking loudly on the door. ”Well,
it looks like our time is up. Spread the word!”
Other scientists are more measured in their estimation of
bonology’s future: ”Seriously? Steve’s sitting in jail right
now? Oh God, what the hell has he gotten himself into?!”
1
1
<β <β+ ,
n
n
and
ε−
1
1
<ε <ε+ .
n
n
Let
1
1
1
1
f (n) = (β − ) − (α − ) − (ω − ) − (ε − ),
n
n
n
n
and let
1
1
1
1
g(n) = (β + ) − (α + ) − (ω + ) − (ε + ).
n
n
n
n
Then we have that, for all n ∈ N,
f (n) < β − α − ω − ε < g(n).
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Taking limits as n → ∞, and noting that both f (n), g(n) → 0
as n → ∞, we arrive at the desired result.
In light of the improved counting methods outlined in
section 3, we may assume the error term is negligible. In this
case, we may apply a similar proof of the above theorem to
conclude that
β = α + ω.
This modified theorem has the following corollary:
Corollary: Suppose that ε = 0; then β = α + ω. In the
case that β and α are given, we may determine the amount ω
of bones left to count in the pile by the formula
ω = β − α.
Similarly, given β and ω, the number α of counted bones is
given by the formula
α = β − ω.
Proof: To show ω = β − α, we proceed by induction on β .
Suppose that β = 1. Then there is exactly one bone in the
pile, which is either counted or uncounted. In the case that it
is counted, then α = 1 and necessarily we must have ω = 0 (a
bone cannot be both counted and uncounted at the same time)
and the equation holds. In the case that the bone is uncounted,
we have that α = 0, hence ω = 1 and the equation holds in
this case as well.
Now, suppose our equation holds for all β ≤ n. For β =
n+1, we use the following trick: imagine we had removed one
bone from our original pile; write β 0 = β − 1 for the number
of bones in this modified pile, with α 0 and ω 0 representing the
number of counted and uncounted bones. We then have β 0 =
β − 1 = (n + 1) − 1 = n. Applying our induction hypothesis,
then, we have
ω 0 = β 0 − α 0.
Now, adding one more bone to the pile represented by β 0 will
either add one to either the total of counted bones or to the
total of uncounted bones (but not both). Thus, we have either
α = α 0 + 1 or ω = ω 0 + 1. Note also that after adding one
bone, we have that β = β 0 + 1. Furthermore, we have that
if α is one more than α 0 , then α 0 is one less than α, hence
α = α 0 + 1 implies α − 1 = α 0 . Similarly, ω = ω 0 + 1 implies
ω − 1 = ω 0 . In the case of α − 1 = α 0 , we have that the
amount of uncounted bones in both our original and modified
piles is the same, i.e. ω 0 = ω. We then rewrite the equation
above as
ω = ω 0 = β 0 − α 0 = (β − 1) − (α + 1) = β − α.
Similarly, in the case that ω − 1 = ω 0 we have α 0 = α, and
thus
ω − 1 = ω 0 = β 0 − α 0 = (β − 1) − α.
Observe that, if the number of counted bones is fixed, and we
subtract one from the total number of bones, then we must
necessarily subtract one from the number of uncounted bones.
Thus, we have
ω = β − α,
completing the induction.
By an identical argument, we may show that α = β − ω,
completing the proof.