Cultural adaptation in mathematics and physics
Abstract The thesis is that the organization, attitudes, and customs
of a scientific discipline are strongly influenced by the nature of the
subject matter. A case study is presented, comparing theoretical physics and pure
mathematics. These share a great deal, but differences in goals and subject
have led to striking cultural differences and a long history of culture
shock at the interface. An analysis of this sort can be useful in developing
science policy and managing change. We conclude for
example that mathematics and physics have different needs in ethical standards, grant support, and electronic publication.
Scientists deliberately adapt technical methods to the subjects they study. Some cultural adaptations are also deliberate, or at least obviously necessary: for example equipment forces a lot of herd activity in some areas ("big science") while others remain more solitary ("small science"). But scientists are typically unaware of most of
their culture. Social scientists studying science are very aware of culture. Unfortunately many of them assert that culture can be studied without understanding content, or even that culture is primary and determines content. Such an approach by its nature abandons realistic hope of serious influence on science or science policy.
In tranquil times there is not much need for awareness of cultural adaptation: the benefits of adaptation push research communities in the right directions without conscious effort. Unfortunately these
are not tranquil times. Support structures, societal expectations, and even the publication and communication infrastructure are all on the verge of radical change. These changes will
sweep away many slowly and painfully acquired cultural nuances. An awareness of culture and adaptation may make it possible reduce the damage.
We present a case study, contrasting pure mathematics and theoretical physics to reveal adaptation in both areas. This is a relatively simple case becase a wide spectrum of differences traces back to a single root cause: reliability. In other cases, particularly in experimental areas, the roots of culture will be more complex.
By "physics" we will mean theoretical work in areas
like high-energy particles, superconductivity, or quantum
mechanics: areas where sophisticated mathematical apparatus is
needed even to organize or interpret real data. By "mathematics"
we mean the "pure" areas in which abstract mathematical apparatus
is developed. The point of excluding experimental physics and
applied mathematics is to obtain relatively homogeneous groups.
They are still richly diverse in their subcultures, but the
commonalities are strong enough permit useful conclusions.
There is a lot of traffic between these areas. Physicists rely on
mathematical work, and many mathematical structures are inspired
by physics. The shared use of mathematical apparatus and long
history of interaction has led to a great deal of shared
language and strong superficial similarities.
Reliability and logic
The key to the differences between these fields seems to be reliability. Mathematics, through logical rigor, can
achieve essentially complete reliability. Information in physics may be excellent
but is never perfect. We remark that the reliability of logic is an "experimental" fact. The Incompleteness Theorem of Godel showed that we cannot prove that correct logic yields
completely reliable conclusions, even if we set aside concerns about the circularity of proving something about proof, or about human fallibility. However as a hypothesis based on experience, the reliability
of logic has been extremely well tested over several thousand years.
To illustrate the significance of reliability, consider the use of proof
by contradiction. In mathematics it is a standard technique to begin
with a dubious assertion and build an elaborate logical structure on
it. At the end something emerges which is known to be false.
The conclusion is that the initial assertion must also be false. The
usefulness of this method depends heavily on the complete
reliability of logic and of the other information used in the
demonstration. Any leakage will cause the method to fail.
The false conclusion at the end may be a consequence of a
flaw in the argument or an ingredient, rather than the
falsehood of the target assertion.
Mathematical customs have adapted to this difference between
perfect and imperfect information. The conclusion of a mathematical
plausibility argument is traditionally called a "conjecture", while the
result of a rigorous argument is called a "theorem." Theorems can be
used without fear in a contradiction argument;
conjectures are a possible source of error.
Physics does not work this way: elaborate logical structures tend to
magnify errors, so are suspect. A direct plausibility argument is
usually more robust than an elaborate logical "proof." Accordingly
the physics culture places a premum on short insightful arguments
(supported by calculation: see the next section), even if wrong in
detail. The "theorem-conjecture" distinction is not particularly
These differences lead to culture shock. In physics conclusions from
plausible argument have first-class status. Mathematicians tend to
conclusions as "conjectures" still needing proofs, or dismiss them as
imprecise. Physicists resent this. Conversely physicists tend to be
disdainful of mathematical rigor as being excessively
compulsive about detail, and mathematicians resent this. But there
good reasons for the values held in both disciplines. The problems come from customs adapted to the subject, not
xenophobia or a power contest.
The effects of these mutual ill-adaptations are not symmetric.
Mathematical practices used
in physics are inefficient or irrelevant, but not harmful. Physical
distinction between conjecture and theorem) used in mathematics
can cause harm: it
jeopardizes standard techniques such as proof by contradiction.
There is widespread
feeling among mathematicians that violating these "truth in
advertising" customs should
be considered misconduct
. This is a behavior which is productive
in one field and
misconduct in another, not because of contingent historical
development of "standard
practice", but because the subjects are different.
Process versus Outcome
The decisive criterion for correctness in theoretical physics is
experimental observation. This focuses attention on outcomes, not
process. This point of
view permeates even internal efforts in theoretical development that
do not make direct
contact with experiment. When a model is developed it is checked
against others believed
to be relevant: special cases, the "classical limit", etc.
In pure mathematics the primary criterion is internal. The reliability
of logic can be rephrased as:
if an argument produces a false conclusion, then it contains
either a logical flaw or
an erroneous hypothesis. Mathematicians have gone to some lengths to ensure
their hypotheses are reliable, so the absence of logical
flaws is a criterion for correctness. In practice it has been very
effective. Attention is focused on the process (avoiding or detecting
flaws), rather than the outcome.
This difference provides further opportunities for friction. A
mathematician can offer work of great
technical power to the physics community and be dismissed as
having no connection to
"reality": no testable outcomes. A physicist can offer work which
reaches a desired goal
by a magnificent leap of intuition, and be criticized for being sloppy.
There are many other ramifications of this difference in focus. For
are more tolerant of apparently pointless exploration, as long as it
conforms to internal
standards of rigor. Physicists tend to be more relaxed about precision
and more judgmental
about significance. These differences cause problems in grant and paper reviews in border areas.
Customs well-adapted to the subject should maximize return on
resource investment. This means approaches seriously out of step
with local customs may be counterproductive in some way.
Alternatively, these customs may reflect adaptation to some
influence other than the subject matter.
As an illustration we consider different levels of rigor expected in
the two fields. Years often pass between an understanding
satisfactory to physicists and a mathematical demonstration.
Is the insistence on rigor a consequence of being sheltered from the demands of the real world? Or is it more efficient in
some way? The history of mathematics reveals a lot of
backsliding, but the predominent trend is toward greater rigor.
Explaining this begins with another fundamental aspect of
mathematics: since it is (usually) right the first time, it is not
discarded. Over time it may become uninteresting or insignificant,
but it does not become incorrect. As a result mathematics is an
In physics (and most of the rest of science)
material must be checked and refined rather than simply accreting, and customs have evolved to support this.
Duplication is tolerated or even encouraged. a
great deal of material is discarded, and there is a strong secondary
literature to record the outcome of the process. These activities use
resources. Mathematics lacks many of these mechanisms: the payoff
for working slowly and getting it right the first time is savings
in the refinement process. In principle the same payoff is available
to physics: if complete reliability were possible then the most
efficient approach would be to seek it even at great sacrifice of
"local" speed. But this is not possible, and an attempt to import this
attitude into physics would be ill-adapted and counterproductive.
This adaptation has produced a vulnerability in mathematics. A
group or individual can
disregard the customary standards and seem to make rapid progress
by working on a
more intuitive level. But the output is unreliable. Mathematics
largely lacks the mechanisms needed to deal with
such material so this causes problems ranging from areas frozen up
for decades, to
unemployable students, to the outright collapse of entire schools of
As a second illustration of efficiency we consider the "fad"
phenomenon in (theoretical) physics. It sometimes happens that an
area becomes fashionable. There is a flurry of publication, with a lot
of duplication. Then most of the participants drop it and go off to the
next hot area. Physics has been criticized for this short attention
span. But this behavior is probably adapted to the subject matter.
First, the goal is development of intuition and understanding, and
this is an effective group activity. Duplication in publication is like
replication of experiments: several intuitions leading to the same
conclusion increase the likelihood that the conclusion is correct. And
after a period the useful limits of speculation are reached, and it is a
better use of resources to move on than to try to squeeze out a bit
more. However if all activity ceased
after a fad then eventually all of physics would become unsuitable
for further development. Different activities continue:
experimentalists test the testable parts. Mathematicians clean
up the logical parts. A few physicists remain to distill the material
into review and survey articles. And after a period of solidification
the area is ready for another round of theoretical development.
Mathematics has occasional fads, but for the
most part it is a long-term solitary activity. The reviewing journal
divides mathematics into roughly 5,000 subtopics, most sparsely
populated. Mathematicians tend to be less mobile between
specialties for many reasons: a greater technical investment is needed for
progress; big groups are seldom more efficient; and duplication is
unnecessary and usually discouraged. These factors tend to drive
mathematicians apart. In consequence the community lacks the customs
evolved in physics to deal with the aftermath of fads. If
mathematicians desert an area no one comes in afterwards to clean up. There is less tradition of review articles: since
the material is already right there is less sifting to do, and less
compression is possible. Shifts of fashion may be an efficient
behavior in physics, but they are not a good model for mathematics.
These considerations also suggest ways grant programs might be fine-tuned to mesh with cultural nuances. The physics group activity is often focused at conferences, while mathematical conferences are more oriented to communicating results. This suggests that mathematical conferences should (on average) be shorter and more frequent, while physics would benefit more from extended "summer institute" formats. Mathematical program officers might watch active areas for quality control problems, and sponsor physics-style review and consolidation activity. This might be a more effective use of resources than supporting the presentation of the newest results.
Papers in pure math and theoretical physics often look similar, treat
the same subjects, and reside in the same library. However
publication customs and uses of the literature are quite different.
Physicists tend not to use the published primary literature. They
work from current information (preprints, personal contacts), and
the secondary literature (review articles, textbooks). The citation
half-life of physics papers is short, and there are jokes about
"write-only" journals that no one reads. Duplication and rediscovery of
previously published material are common. In contrast many
mathematicians make extensive use of the literature, and in classical
areas it is common to find citations of very old papers.
There are differences in the construction of the literature as well. In
mathematics the refereeing process is usually taken seriously. Errors
tend to get caught, and detailed comments often lead to helpful
revision of the paper. In physics the
peer-review process has low
credibility. Reviewers are uninterested, and their reports do not
carry much weight with either authors or editors. Published papers
are almost always identical to the preprint version.
One view of these differences is that the mathematical primary
literature is user-oriented: genuinely useful to readers. In physics
it is more author-oriented, serving largely to
record the accomplishments of writers.
These differences again reflect differences in the subject matter. The
theoretical physics primary literature is not reliable enough to make
searching it very fruitful. It records the knowledge development
process rather than the end result. If material is incorporated into
the secondary literature or some shared tradition then it is
reasonably accessible, but it is often more efficient to rediscover
something than to sift the primary literature. A consequence is that
there is not much benefit in careful editing or refereeing. This
leads to journals that are, in the words of one mathematician, "like a
blackboard that must periodically be erased." In contrast, the
mathematical literature is reliable enough to be a valuable asset to
The differences in literatures have led to differences in social
structure. As noted above, mathematics has many sparsely populated
specialties. More accurately these could be described as larger
communities distributed in time and communicating through the
literature. This works even though the communication is
because the material is reliable. Less reliable material requires
two-way give and take. As a consequence working groups in physics are
more constricted in time, and appear larger because they are all
visible at once. This also works the other way: a large working group
with a lot of real-time interaction weakens the benefits of reliability, and in fact larger groups in mathematics often do become more casual about
quality control. This in turn leads to a curious problem in the mathematical infrastructure. The leadership in the professional societies and top journals tends to come from larger and more active areas. As a result they tend to underestimate the importance of quality control to the community as a whole.
This analysis has applications to the structuring of
electronic communications. Both mathematicians and physicists have
become heavy users of electronic mail and preprint databases.
Theoretical high-energy physics is particularly advanced due to the leadership of Paul Ginsparg at Los Alamos, and in
that area the current published literature has become nearly
irrelevant. If paper journals perish as a result, readers will
lose the quality control, and authors will lose some credit
mechanisms. In this area the quality control is marginal, and
seems a small price to pay for the greatly increased speed and
functionality. Authors may be briefly inconvenienced but new
recognition mechanisms are already developing.
The needs of mathematics are different. If the physics example were
followed too closely, and led to a significant decline in reliability, it
would yield a literature seriously out of step with the needs of the
Sociological symptoms might include the demise of sparsely populated
areas and an increase
in size of working groups. New quality-control mechanisms would
eventually evolve, but
these would take time and are likely to be less satisfactory than the
literature-wide control. Re-adaptation of the social structure might
take quite a long
time. A mathematics-specific electronic publication model with
greater emphasis on quality
control seems to be called for .
There are strong outside influences on science. Some are accidental
byproducts of other circumstances. For example the current
underrepresentation of some racial groups and genders surely results from social forces unrelated to science. A more subtle example is given by Harwood , who
argues that the old German ideal of the "universal scholar"
led to a larger proportion of "mandarins" to "pragmatists" in early
German genetics, as compared to the United States.
The more interesting influences are ones purposefully directed at
science. There are inappropriate and obviously counterproductive
examples like Russian genetics in the Lysenko era,
Ayrian science in Nazi Germany, or church-controlled astronomy in Galileo's time. Some
influences are appropriate in principle: society can reasonably expect
some return on the investment, and is
entitled to push science in productive directions. However these
influences can still interfere with adaptation to the subject, and can be
counterproductive. For instance Montgomery 
suggests that plasma physics has been harmed by the forced march
toward fusion. For a more subtle problem we note that it has been NSF policy for some
years to encourage mathematicians to use computers. This is straightforward in applied work. It is harder to organize computation to provide the reliability crucial to pure mathematics. As a result quite a few mathematicians who wanted to use machines for more than e-mail and word processing have moved to applied areas. The machine/pencil dichotomy also seems to attract students to applied work. This shift of the entire field was probably not an intended consequence of the original program.
Many areas of science have been unusually free of societal pressure
in the last fifty years: in the U. S. Vannevar Bush's "social contract"
led to uncritical support of science
as an abstract public good. In the Soviet Union it was often regarded
as "production" and therefore intrinsically good. This era is
. As science policy becomes more
demanding there is an increasing urgency to design it to mesh with the
cultural structures preferred by nature.
We have presented an analysis of the adaptation of culture and
custom to subject matter in two scientific fields. Understanding the
differences leads to conclusions about interdisciplinary work,
professional ethics, electronic communications, science policy, and
other "infrastructure" issues.
There are several cautions. First, this should be seen as
explanations for observed cultural differences, not "proofs"
that they must exist. Second, not all cultural differences are related
to subject matter. Differences can come from societal influences, as
discussed above, or from things like "founder effects" where
personalities or circumstances of the formation of the field have left
a lingering imprint. A final caution concerns the drawing of
boundaries. Micro-adaptation produces
cultural variation on fine scales. For example in experimental biology
adaptations of researcher and organism to each another that drive
diversity on a very
small scale . Consequently the strongest conclusions are limited
to small scales and
larger scale comparisons are limited to commonalties that transcend
local variations. Pure
mathematics and theoretical physics have significantly different
commonalities, but if we
had included applied mathematics and experimental physics then
internal diversity would
have overwhelmed the differences.
There are conclusions at the largest scale --- all of science --- just
from the fact that there are cultural differences. Interdisciplinary
workers should respect other cultures; there can be no detailed uniform code
of ethics; science policy should mesh with cultural adaptations;
area-specific nuances of publication should be
preserved in the transition to electronic formats. Generally, a
one-size-fits-all approach to any infrastructure issue will be sub-optimal.
Finally, there are many other cultural divides in science resulting from
differences in subject matter. There is the large science -- small
science division mentioned in the introduction. Some areas (the
genome project, x-ray crystallography) are primarily oriented to the
production and analysis of data, while others (mathematics,
theoretical physics) consist almost entirely of discursive argument.
Subjects that require sophisticated use of statistics can be expected
to differ from those that do not. Subjects like evolutionary biology
and astronomy are oriented toward explanatory stories that organize
observation, while the classical laboratory sciences emphasize
testable prediction. Purely academic subjects differ from ones with
significant commercial or national security interest. A great deal of
worthwhile information should result from analysis of cultural
adaptations to these differences.
Background for this article appears in: A. Jaffe and
F. Quinn, Theoretical mathematics: a cultural synthesis of
mathematics and theoretical Physics . Bulletin of
the American Math. Soc. Vol. 29 (1993) pp. 1--13.
Not all mathematicians agree. For example W.~Thurston (
On proof and progress in mathematics, Bull. Am. Math. Soc. 30
(1994) pp. 161--177) suggests the primary goal of mathematicians
should be human understanding of mathematics, rather than
the production of reliable knowledge. On this basis he argues for
heuristic argument as a basis for mathematical knowledge. This is,
however, very much a minority view.
See Theoretical mathematics, note~1.
This issue is explored in more detail in F. Quinn, Roadkill on the electronic highway: The
threat to the mathematical literature Notices of the Amer. Math Soc. 42 (1995) 53-56.
Jonathan Harwood Styles of Scientific Thought: the German
Genetics Community 1900--1933 University of Chicago Press, 1993.
David Montgomery, letter to Science, vol. 269 (1995)
R. Byerly and R. A. Pielke, The changing ecology of
United States Science, Science 269 (1995), pp. 1531--2.
See papers in the collection "The right organism for the job",
J. History of Biology 26 (1993) 233--368.