The scientific method is an empirical method
of acquiring knowledge that has characterized
the development of science since at least
the 17th century.
It involves careful observation, applying
rigorous skepticism about what is observed,
given that cognitive assumptions can distort
how one interprets the observation.
It involves formulating hypotheses, via induction,
based on such observations; experimental and
measurement-based testing of deductions drawn
from the hypotheses; and refinement (or elimination)
of the hypotheses based on the experimental
findings.
These are principles of the scientific method,
as distinguished from a definitive series
of steps applicable to all scientific enterprises.Though
diverse models for the scientific method are
available, there is in general a continuous
process that includes observations about the
natural world.
People are naturally inquisitive, so they
often come up with questions about things
they see or hear, and they often develop ideas
or hypotheses about why things are the way
they are.
The best hypotheses lead to predictions that
can be tested in various ways.
The most conclusive testing of hypotheses
comes from reasoning based on carefully controlled
experimental data.
Depending on how well additional tests match
the predictions, the original hypothesis may
require refinement, alteration, expansion
or even rejection.
If a particular hypothesis becomes very well
supported, a general theory may be developed.Although
procedures vary from one field of inquiry
to another, they are frequently the same from
one to another.
The process of the scientific method involves
making conjectures (hypotheses), deriving
predictions from them as logical consequences,
and then carrying out experiments or empirical
observations based on those predictions.
A hypothesis is a conjecture, based on knowledge
obtained while seeking answers to the question.
The hypothesis might be very specific, or
it might be broad.
Scientists then test hypotheses by conducting
experiments or studies.
A scientific hypothesis must be falsifiable,
implying that it is possible to identify a
possible outcome of an experiment or observation
that conflicts with predictions deduced from
the hypothesis; otherwise, the hypothesis
cannot be meaningfully tested.The purpose
of an experiment is to determine whether observations
agree with or conflict with the predictions
derived from a hypothesis.
Experiments can take place anywhere from a
garage to CERN's Large Hadron Collider.
There are difficulties in a formulaic statement
of method, however.
Though the scientific method is often presented
as a fixed sequence of steps, it represents
rather a set of general principles.
Not all steps take place in every scientific
inquiry (nor to the same degree), and they
are not always in the same order.
Some philosophers and scientists have argued
that there is no scientific method; they include
physicist Lee Smolin and philosopher Paul
Feyerabend (in his Against Method).
Robert Nola and Howard Sankey remark that
"For some, the whole idea of a theory of scientific
method is yester-year's debate, the continuation
of which can be summed up as yet more of the
proverbial deceased equine castigation.
We beg to differ."
== 
History ==
Important debates in the history of science
concern rationalism, especially as advocated
by René Descartes; inductivism and/or empiricism,
as argued for by Francis Bacon, and rising
to particular prominence with Isaac Newton
and his followers; and hypothetico-deductivism,
which came to the fore in the early 19th century.
The term "scientific method" emerged in the
19th century, when a significant institutional
development of science was taking place and
terminologies establishing clear boundaries
between science and non-science, such as "scientist"
and "pseudoscience", appeared.
Throughout the 1830s and 1850s, by which time
Baconianism was popular, naturalists like
William Whewell, John Herschel, John Stuart
Mill engaged in debates over "induction" and
"facts" and were focused on how to generate
knowledge.
In the late 19th and early 20th centuries,
a debate over realism vs. antirealism was
conducted as powerful scientific theories
extended beyond the realm of the observable.The
term "scientific method" came into popular
use in the twentieth century, popping up in
dictionaries and science textbooks, although
there was little scientific consensus over
its meaning.
Although there was a growth through the middle
of the twentieth century, by the end of that
century numerous influential philosophers
of science like Thomas Kuhn and Paul Feyerabend
had questioned the universality of the "scientific
method" and in doing so largely replaced the
notion of science as a homogeneous and universal
method with that of it being a heterogeneous
and local practice.
In particular, Paul Feyerabend argued against
there being any universal rules of science.
Historian of science Daniel Thurs maintains
that the scientific method is a myth or, at
best, an idealization.
== Overview ==
The scientific method is the process by which
science is carried out.
As in other areas of inquiry, science (through
the scientific method) can build on previous
knowledge and develop a more sophisticated
understanding of its topics of study over
time.
This model can be seen to underlie the scientific
revolution.The ubiquitous element in the model
of the scientific method is empiricism, or
more precisely, epistemologic sensualism.
This is in opposition to stringent forms of
rationalism: the scientific method embodies
that reason alone cannot solve a particular
scientific problem.
A strong formulation of the scientific method
is not always aligned with a form of empiricism
in which the empirical data is put forward
in the form of experience or other abstracted
forms of knowledge; in current scientific
practice, however, the use of scientific modelling
and reliance on abstract typologies and theories
is normally accepted.
The scientific method is of necessity also
an expression of an opposition to claims that
e.g. revelation, political or religious dogma,
appeals to tradition, commonly held beliefs,
common sense, or, importantly, currently held
theories, are the only possible means of demonstrating
truth.
Different early expressions of empiricism
and the scientific method can be found throughout
history, for instance with the ancient Stoics,
Epicurus, Alhazen, Roger Bacon, and William
of Ockham.
From the 16th century onwards, experiments
were advocated by Francis Bacon, and performed
by Giambattista della Porta, Johannes Kepler,
and Galileo Galilei.
There was particular development aided by
theoretical works by Francisco Sanches, John
Locke, George Berkeley, and David Hume.
The hypothetico-deductive model formulated
in the 20th century, is the ideal although
it has undergone significant revision since
first proposed (for a more formal discussion,
see below).
Staddon (2017) argues it is a mistake to try
following rules which are best learned through
careful study of examples of scientific investigation.
=== Process ===
The overall process involves making conjectures
(hypotheses), deriving predictions from them
as logical consequences, and then carrying
out experiments based on those predictions
to determine whether the original conjecture
was correct.
There are difficulties in a formulaic statement
of method, however.
Though the scientific method is often presented
as a fixed sequence of steps, these actions
are better considered as general principles.
Not all steps take place in every scientific
inquiry (nor to the same degree), and they
are not always done in the same order.
As noted by scientist and philosopher William
Whewell (1794–1866), "invention, sagacity,
[and] genius" are required at every step.
==== Formulation of a question ====
The question can refer to the explanation
of a specific observation, as in "Why is the
sky blue?" but can also be open-ended, as
in "How can I design a drug to cure this particular
disease?"
This stage frequently involves finding and
evaluating evidence from previous experiments,
personal scientific observations or assertions,
as well as the work of other scientists.
If the answer is already known, a different
question that builds on the evidence can be
posed.
When applying the scientific method to research,
determining a good question can be very difficult
and it will affect the outcome of the investigation.
==== Hypothesis ====
A hypothesis is a conjecture, based on knowledge
obtained while formulating the question, that
may explain any given behavior.
The hypothesis might be very specific; for
example, Einstein's equivalence principle
or Francis Crick's "DNA makes RNA makes protein",
or it might be broad; for example, unknown
species of life dwell in the unexplored depths
of the oceans.
A statistical hypothesis is a conjecture about
a given statistical population.
For example, the population might be people
with a particular disease.
The conjecture might be that a new drug will
cure the disease in some of those people.
Terms commonly associated with statistical
hypotheses are null hypothesis and alternative
hypothesis.
A null hypothesis is the conjecture that the
statistical hypothesis is false; for example,
that the new drug does nothing and that any
cure is caused by chance.
Researchers normally want to show that the
null hypothesis is false.
The alternative hypothesis is the desired
outcome, that the drug does better than chance.
A final point: a scientific hypothesis must
be falsifiable, meaning that one can identify
a possible outcome of an experiment that conflicts
with predictions deduced from the hypothesis;
otherwise, it cannot be meaningfully tested.
==== Prediction ====
This step involves determining the logical
consequences of the hypothesis.
One or more predictions are then selected
for further testing.
The more unlikely that a prediction would
be correct simply by coincidence, then the
more convincing it would be if the prediction
were fulfilled; evidence is also stronger
if the answer to the prediction is not already
known, due to the effects of hindsight bias
(see also postdiction).
Ideally, the prediction must also distinguish
the hypothesis from likely alternatives; if
two hypotheses make the same prediction, observing
the prediction to be correct is not evidence
for either one over the other.
(These statements about the relative strength
of evidence can be mathematically derived
using Bayes' Theorem).
==== Testing ====
This is an investigation of whether the real
world behaves as predicted by the hypothesis.
Scientists (and other people) test hypotheses
by conducting experiments.
The purpose of an experiment is to determine
whether observations of the real world agree
with or conflict with the predictions derived
from a hypothesis.
If they agree, confidence in the hypothesis
increases; otherwise, it decreases.
Agreement does not assure that the hypothesis
is true; future experiments may reveal problems.
Karl Popper advised scientists to try to falsify
hypotheses, i.e., to search for and test those
experiments that seem most doubtful.
Large numbers of successful confirmations
are not convincing if they arise from experiments
that avoid risk.
Experiments should be designed to minimize
possible errors, especially through the use
of appropriate scientific controls.
For example, tests of medical treatments are
commonly run as double-blind tests.
Test personnel, who might unwittingly reveal
to test subjects which samples are the desired
test drugs and which are placebos, are kept
ignorant of which are which.
Such hints can bias the responses of the test
subjects.
Furthermore, failure of an experiment does
not necessarily mean the hypothesis is false.
Experiments always depend on several hypotheses,
e.g., that the test equipment is working properly,
and a failure may be a failure of one of the
auxiliary hypotheses.
(See the Duhem–Quine thesis.)
Experiments can be conducted in a college
lab, on a kitchen table, at CERN's Large Hadron
Collider, at the bottom of an ocean, on Mars
(using one of the working rovers), and so
on.
Astronomers do experiments, searching for
planets around distant stars.
Finally, most individual experiments address
highly specific topics for reasons of practicality.
As a result, evidence about broader topics
is usually accumulated gradually.
==== Analysis ====
This involves determining what the results
of the experiment show and deciding on the
next actions to take.
The predictions of the hypothesis are compared
to those of the null hypothesis, to determine
which is better able to explain the data.
In cases where an experiment is repeated many
times, a statistical analysis such as a chi-squared
test may be required.
If the evidence has falsified the hypothesis,
a new hypothesis is required; if the experiment
supports the hypothesis but the evidence is
not strong enough for high confidence, other
predictions from the hypothesis must be tested.
Once a hypothesis is strongly supported by
evidence, a new question can be asked to provide
further insight on the same topic.
Evidence from other scientists and experience
are frequently incorporated at any stage in
the process.
Depending on the complexity of the experiment,
many iterations may be required to gather
sufficient evidence to answer a question with
confidence, or to build up many answers to
highly specific questions in order to answer
a single broader question.
=== DNA example ===
The basic elements of the scientific method
are illustrated by the following example from
the discovery of the structure of DNA:
Question: Previous investigation of DNA had
determined its chemical composition (the four
nucleotides), the structure of each individual
nucleotide, and other properties.
It had been identified as the carrier of genetic
information by the Avery–MacLeod–McCarty
experiment in 1944, but the mechanism of how
genetic information was stored in DNA was
unclear.
Hypothesis: Linus Pauling, Francis Crick and
James D. Watson hypothesized that DNA had
a helical structure.
Prediction: If DNA had a helical structure,
its X-ray diffraction pattern would be X-shaped.
This prediction was determined using the mathematics
of the helix transform, which had been derived
by Cochran, Crick and Vand (and independently
by Stokes).
This prediction was a mathematical construct,
completely independent from the biological
problem at hand.
Experiment: Rosalind Franklin crystallized
pure DNA and performed X-ray diffraction to
produce photo 51.
The results showed an X-shape.
Analysis: When Watson saw the detailed diffraction
pattern, he immediately recognized it as a
helix.
He and Crick then produced their model, using
this information along with the previously
known information about DNA's composition
and about molecular interactions such as hydrogen
bonds.The discovery became the starting point
for many further studies involving the genetic
material, such as the field of molecular genetics,
and it was awarded the Nobel Prize in 1962.
Each step of the example is examined in more
detail later in the article.
=== Other components ===
The scientific method also includes other
components required even when all the iterations
of the steps above have been completed:
==== Replication ====
If an experiment cannot be repeated to produce
the same results, this implies that the original
results might have been in error.
As a result, it is common for a single experiment
to be performed multiple times, especially
when there are uncontrolled variables or other
indications of experimental error.
For significant or surprising results, other
scientists may also attempt to replicate the
results for themselves, especially if those
results would be important to their own work.
==== External review ====
The process of peer review involves evaluation
of the experiment by experts, who typically
give their opinions anonymously.
Some journals request that the experimenter
provide lists of possible peer reviewers,
especially if the field is highly specialized.
Peer review does not certify correctness of
the results, only that, in the opinion of
the reviewer, the experiments themselves were
sound (based on the description supplied by
the experimenter).
If the work passes peer review, which occasionally
may require new experiments requested by the
reviewers, it will be published in a peer-reviewed
scientific journal.
The specific journal that publishes the results
indicates the perceived quality of the work.
==== Data recording and sharing ====
Scientists typically are careful in recording
their data, a requirement promoted by Ludwik
Fleck (1896–1961) and others.
Though not typically required, they might
be requested to supply this data to other
scientists who wish to replicate their original
results (or parts of their original results),
extending to the sharing of any experimental
samples that may be difficult to obtain.
== Scientific inquiry ==
Scientific inquiry generally aims to obtain
knowledge in the form of testable explanations
that scientists can use to
predict the results of future experiments.
This allows scientists to gain a better understanding
of the topic under study, and later to use
that understanding to intervene in its causal
mechanisms (such as to cure disease).
The better an explanation is at making predictions,
the more useful it frequently can be, and
the more likely it will continue to explain
a body of evidence better than its alternatives.
The most successful explanations – those
which explain and make accurate predictions
in a wide range of circumstances – are often
called scientific theories.
Most experimental results do not produce large
changes in human understanding; improvements
in theoretical scientific understanding typically
result from a gradual process of development
over time, sometimes across different domains
of science.
Scientific models vary in the extent to which
they have been experimentally tested and for
how long, and in their acceptance in the scientific
community.
In general, explanations become accepted over
time as evidence accumulates on a given topic,
and the explanation in question proves more
powerful than its alternatives at explaining
the evidence.
Often subsequent researchers re-formulate
the explanations over time, or combined explanations
to produce new explanations.
Tow sees the scientific method in terms of
an evolutionary algorithm applied to science
and technology.
=== Properties of scientific inquiry ===
Scientific knowledge is closely tied to empirical
findings, and can remain subject to falsification
if new experimental observation incompatible
with it is found.
That is, no theory can ever be considered
final, since new problematic evidence might
be discovered.
If such evidence is found, a new theory may
be proposed, or (more commonly) it is found
that modifications to the previous theory
are sufficient to explain the new evidence.
The strength of a theory can be argued to
relate to how long it has persisted without
major alteration to its core principles.
Theories can also become subsumed by other
theories.
For example, Newton's laws explained thousands
of years of scientific observations of the
planets almost perfectly.
However, these laws were then determined to
be special cases of a more general theory
(relativity), which explained both the (previously
unexplained) exceptions to Newton's laws and
predicted and explained other observations
such as the deflection of light by gravity.
Thus, in certain cases independent, unconnected,
scientific observations can be connected to
each other, unified by principles of increasing
explanatory power.Since new theories might
be more comprehensive than what preceded them,
and thus be able to explain more than previous
ones, successor theories might be able to
meet a higher standard by explaining a larger
body of observations than their predecessors.
For example, the theory of evolution explains
the diversity of life on Earth, how species
adapt to their environments, and many other
patterns observed in the natural world; its
most recent major modification was unification
with genetics to form the modern evolutionary
synthesis.
In subsequent modifications, it has also subsumed
aspects of many other fields such as biochemistry
and molecular biology.
=== Beliefs and biases ===
Scientific methodology often directs that
hypotheses be tested in controlled conditions
wherever possible.
This is frequently possible in certain areas,
such as in the biological sciences, and more
difficult in other areas, such as in astronomy.
The practice of experimental control and reproducibility
can have the effect of diminishing the potentially
harmful effects of circumstance, and to a
degree, personal bias.
For example, pre-existing beliefs can alter
the interpretation of results, as in confirmation
bias; this is a heuristic that leads a person
with a particular belief to see things as
reinforcing their belief, even if another
observer might disagree (in other words, people
tend to observe what they expect to observe).
A historical example is the belief that the
legs of a galloping horse are splayed at the
point when none of the horse's legs touches
the ground, to the point of this image being
included in paintings by its supporters.
However, the first stop-action pictures of
a horse's gallop by Eadweard Muybridge showed
this to be false, and that the legs are instead
gathered together.Another important human
bias that plays a role is a preference for
new, surprising statements (see appeal to
novelty), which can result in a search for
evidence that the new is true.
Poorly attested beliefs can be believed and
acted upon via a less rigorous heuristic.Goldhaber
and Nieto published in 2010 the observation
that if theoretical structures with "many
closely neighboring subjects are described
by connecting theoretical concepts then the
theoretical structure .. becomes increasingly
hard to overturn".
When a narrative is constructed its elements
become easier to believe.
For more on the narrative fallacy, see also
Fleck 1979, p. 27: "Words and ideas are originally
phonetic and mental equivalences of the experiences
coinciding with them.
... Such proto-ideas are at first always too
broad and insufficiently specialized.
... Once a structurally complete and closed
system of opinions consisting of many details
and relations has been formed, it offers enduring
resistance to anything that contradicts it."
Sometimes, these have their elements assumed
a priori, or contain some other logical or
methodological flaw in the process that ultimately
produced them.
Donald M. MacKay has analyzed these elements
in terms of limits to the accuracy of measurement
and has related them to instrumental elements
in a category of measurement.
== Elements of the scientific method ==
There are different ways of outlining the
basic method used for scientific inquiry.
The scientific community and philosophers
of science generally agree on the following
classification of method components.
These methodological elements and organization
of procedures tend to be more characteristic
of natural sciences than social sciences.
Nonetheless, the cycle of formulating hypotheses,
testing and analyzing the results, and formulating
new hypotheses, will resemble the cycle described
below.
The scientific method is an iterative, cyclical
process through which information is continually
revised.
It is generally recognized to develop advances
in knowledge through the following elements,
in varying combinations or contributions:
Characterizations (observations, definitions,
and measurements of the subject of inquiry)
Hypotheses (theoretical, hypothetical explanations
of observations and measurements of the subject)
Predictions (inductive and deductive reasoning
from the hypothesis or theory)
Experiments (tests of all of the above)Each
element of the scientific method is subject
to peer review for possible mistakes.
These activities do not describe all that
scientists do (see below) but apply mostly
to experimental sciences (e.g., physics, chemistry,
and biology).
The elements above are often taught in the
educational system as "the scientific method".The
scientific method is not a single recipe:
it requires intelligence, imagination, and
creativity.
In this sense, it is not a mindless set of
standards and procedures to follow,
but is rather an ongoing cycle, constantly
developing more useful, accurate and comprehensive
models and methods.
For example, when Einstein developed the Special
and General Theories of Relativity, he did
not in any way refute or discount Newton's
Principia.
On the contrary, if the astronomically large,
the vanishingly small, and the extremely fast
are removed from Einstein's theories – all
phenomena Newton could not have observed – Newton's
equations are what remain.
Einstein's theories are expansions and refinements
of Newton's theories and, thus, increase confidence
in Newton's work.
A linearized, pragmatic scheme of the four
points above is sometimes offered as a guideline
for proceeding:
Define a question
Gather information and resources (observe)
Form an explanatory hypothesis
Test the hypothesis by performing an experiment
and collecting data in a reproducible manner
Analyze the data
Interpret the data and draw conclusions that
serve as a starting point for new hypothesis
Publish results
Retest (frequently done by other scientists)The
iterative cycle inherent in this step-by-step
method goes from point 3 to 6 back to 3 again.
While this schema outlines a typical hypothesis/testing
method, it should also be noted that a number
of philosophers, historians, and sociologists
of science, including Paul Feyerabend, claim
that such descriptions of scientific method
have little relation to the ways that science
is actually practiced.
=== Characterizations ===
The scientific method depends upon increasingly
sophisticated characterizations of the subjects
of investigation.
(The subjects can also be called unsolved
problems or the unknowns.)
For example, Benjamin Franklin conjectured,
correctly, that St. Elmo's fire was electrical
in nature, but it has taken a long series
of experiments and theoretical changes to
establish this.
While seeking the pertinent properties of
the subjects, careful thought may also entail
some definitions and observations; the observations
often demand careful measurements and/or counting.
The systematic, careful collection of measurements
or counts of relevant quantities is often
the critical difference between pseudo-sciences,
such as alchemy, and science, such as chemistry
or biology.
Scientific measurements are usually tabulated,
graphed, or mapped, and statistical manipulations,
such as correlation and regression, performed
on them.
The measurements might be made in a controlled
setting, such as a laboratory, or made on
more or less inaccessible or unmanipulatable
objects such as stars or human populations.
The measurements often require specialized
scientific instruments such as thermometers,
spectroscopes, particle accelerators, or voltmeters,
and the progress of a scientific field is
usually intimately tied to their invention
and improvement.
I am not accustomed to saying anything with
certainty after only one or two observations.
==== Uncertainty ====
Measurements in scientific work are also usually
accompanied by estimates of their uncertainty.
The uncertainty is often estimated by making
repeated measurements of the desired quantity.
Uncertainties may also be calculated by consideration
of the uncertainties of the individual underlying
quantities used.
Counts of things, such as the number of people
in a nation at a particular time, may also
have an uncertainty due to data collection
limitations.
Or counts may represent a sample of desired
quantities, with an uncertainty that depends
upon the sampling method used and the number
of samples taken.
==== Definition ====
Measurements demand the use of operational
definitions of relevant quantities.
That is, a scientific quantity is described
or defined by how it is measured, as opposed
to some more vague, inexact or "idealized"
definition.
For example, electric current, measured in
amperes, may be operationally defined in terms
of the mass of silver deposited in a certain
time on an electrode in an electrochemical
device that is described in some detail.
The operational definition of a thing often
relies on comparisons with standards: the
operational definition of "mass" ultimately
relies on the use of an artifact, such as
a particular kilogram of platinum-iridium
kept in a laboratory in France.
The scientific definition of a term sometimes
differs substantially from its natural language
usage.
For example, mass and weight overlap in meaning
in common discourse, but have distinct meanings
in mechanics.
Scientific quantities are often characterized
by their units of measure which can later
be described in terms of conventional physical
units when communicating the work.
New theories are sometimes developed after
realizing certain terms have not previously
been sufficiently clearly defined.
For example, Albert Einstein's first paper
on relativity begins by defining simultaneity
and the means for determining length.
These ideas were skipped over by Isaac Newton
with, "I do not define time, space, place
and motion, as being well known to all."
Einstein's paper then demonstrates that they
(viz., absolute time and length independent
of motion) were approximations.
Francis Crick cautions us that when characterizing
a subject, however, it can be premature to
define something when it remains ill-understood.
In Crick's study of consciousness, he actually
found it easier to study awareness in the
visual system, rather than to study free will,
for example.
His cautionary example was the gene; the gene
was much more poorly understood before Watson
and Crick's pioneering discovery of the structure
of DNA; it would have been counterproductive
to spend much time on the definition of the
gene, before them.
==== DNA-characterizations ====
The history of the discovery of the structure
of DNA is a classic example of the elements
of the scientific method: in 1950 it was known
that genetic inheritance had a mathematical
description, starting with the studies of
Gregor Mendel, and that DNA contained genetic
information (Oswald Avery's transforming principle).
But the mechanism of storing genetic information
(i.e., genes) in DNA was unclear.
Researchers in Bragg's laboratory at Cambridge
University made X-ray diffraction pictures
of various molecules, starting with crystals
of salt, and proceeding to more complicated
substances.
Using clues painstakingly assembled over decades,
beginning with its chemical composition, it
was determined that it should be possible
to characterize the physical structure of
DNA, and the X-ray images would be the vehicle.
..2.
DNA-hypotheses
==== Another example: precession of Mercury
====
The characterization element can require extended
and extensive study, even centuries.
It took thousands of years of measurements,
from the Chaldean, Indian, Persian, Greek,
Arabic and European astronomers, to fully
record the motion of planet Earth.
Newton was able to include those measurements
into consequences of his laws of motion.
But the perihelion of the planet Mercury's
orbit exhibits a precession that cannot be
fully explained by Newton's laws of motion
(see diagram to the right), as Leverrier pointed
out in 1859.
The observed difference for Mercury's precession
between Newtonian theory and observation was
one of the things that occurred to Albert
Einstein as a possible early test of his theory
of General relativity.
His relativistic calculations matched observation
much more closely than did Newtonian theory.
The difference is approximately 43 arc-seconds
per century.
=== Hypothesis development ===
A hypothesis is a suggested explanation of
a phenomenon, or alternately a reasoned proposal
suggesting a possible correlation between
or among a set of phenomena.
Normally hypotheses have the form of a mathematical
model.
Sometimes, but not always, they can also be
formulated as existential statements, stating
that some particular instance of the phenomenon
being studied has some characteristic and
causal explanations, which have the general
form of universal statements, stating that
every instance of the phenomenon has a particular
characteristic.
Scientists are free to use whatever resources
they have – their own creativity, ideas
from other fields, inductive reasoning, Bayesian
inference, and so on – to imagine possible
explanations for a phenomenon under study.
Albert Einstein once observed that "there
is no logical bridge between phenomena and
their theoretical principles."
Charles Sanders Peirce, borrowing a page from
Aristotle (Prior Analytics, 2.25) described
the incipient stages of inquiry, instigated
by the "irritation of doubt" to venture a
plausible guess, as abductive reasoning.
The history of science is filled with stories
of scientists claiming a "flash of inspiration",
or a hunch, which then motivated them to look
for evidence to support or refute their idea.
Michael Polanyi made such creativity the centerpiece
of his discussion of methodology.
William Glen observes that
the success of a hypothesis, or its service
to science, lies not simply in its perceived
"truth", or power to displace, subsume or
reduce a predecessor idea, but perhaps more
in its ability to stimulate the research that
will illuminate ... bald suppositions and
areas of vagueness.
In general scientists tend to look for theories
that are "elegant" or "beautiful".
In contrast to the usual English use of these
terms, they here refer to a theory in accordance
with the known facts, which is nevertheless
relatively simple and easy to handle.
Occam's Razor serves as a rule of thumb for
choosing the most desirable amongst a group
of equally explanatory hypotheses.
To minimize the confirmation bias which results
from entertaining a single hypothesis, strong
inference emphasizes the need for entertaining
multiple alternative hypotheses.
==== DNA-hypotheses ====
Linus Pauling proposed that DNA might be a
triple helix.
This hypothesis was also considered by Francis
Crick and James D. Watson but discarded.
When Watson and Crick learned of Pauling's
hypothesis, they understood from existing
data that Pauling was wrong and that Pauling
would soon admit his difficulties with that
structure.
So, the race was on to figure out the correct
structure (except that Pauling did not realize
at the time that he was in a race) ..3.
DNA-predictions
=== Predictions 
from the hypothesis ===
Any useful hypothesis will enable predictions,
by reasoning including deductive reasoning.
It might predict the outcome of an experiment
in a laboratory setting or the observation
of a phenomenon in nature.
The prediction can also be statistical and
deal only with probabilities.
It is essential that the outcome of testing
such a prediction be currently unknown.
Only in this case does a successful outcome
increase the probability that the hypothesis
is true.
If the outcome is already known, it is called
a consequence and should have already been
considered while formulating the hypothesis.
If the predictions are not accessible by observation
or experience, the hypothesis is not yet testable
and so will remain to that extent unscientific
in a strict sense.
A new technology or theory might make the
necessary experiments feasible.
For example, while a hypothesis on the existence
of other intelligent species may be convincing
with scientifically based speculation, there
is no known experiment that can test this
hypothesis.
Therefore, science itself can have little
to say about the possibility.
In future, a new technique may allow for an
experimental test and the speculation would
then become part of accepted science.
==== DNA-predictions ====
James D. Watson, Francis Crick, and others
hypothesized that DNA had a helical structure.
This implied that DNA's X-ray diffraction
pattern would be 'x shaped'.
This prediction followed from the work of
Cochran, Crick and Vand (and independently
by Stokes).
The Cochran-Crick-Vand-Stokes theorem provided
a mathematical explanation for the empirical
observation that diffraction from helical
structures produces x shaped patterns.
In their first paper, Watson and Crick also
noted that the double helix structure they
proposed provided a simple mechanism for DNA
replication, writing, "It has not escaped
our notice that the specific pairing we have
postulated immediately suggests a possible
copying mechanism for the genetic material".
..4.
DNA-experiments
==== Another example: general relativity ====
Einstein's theory of General Relativity makes
several specific predictions about the observable
structure of space-time, such as that light
bends in a gravitational field, and that the
amount of bending depends in a precise way
on the strength of that gravitational field.
Arthur Eddington's observations made during
a 1919 solar eclipse supported General Relativity
rather than Newtonian gravitation.
=== Experiments ===
Once predictions are made, they can be sought
by experiments.
If the test results contradict the predictions,
the hypotheses which entailed them are called
into question and become less tenable.
Sometimes the experiments are conducted incorrectly
or are not very well designed, when compared
to a crucial experiment.
If the experimental results confirm the predictions,
then the hypotheses are considered more likely
to be correct, but might still be wrong and
continue to be subject to further testing.
The experimental control is a technique for
dealing with observational error.
This technique uses the contrast between multiple
samples (or observations) under differing
conditions to see what varies or what remains
the same.
We vary the conditions for each measurement,
to help isolate what has changed.
Mill's canons can then help us figure out
what the important factor is.
Factor analysis is one technique for discovering
the important factor in an effect.
Depending on the predictions, the experiments
can have different shapes.
It could be a classical experiment in a laboratory
setting, a double-blind study or an archaeological
excavation.
Even taking a plane from New York to Paris
is an experiment which tests the aerodynamical
hypotheses used for constructing the plane.
Scientists assume an attitude of openness
and accountability on the part of those conducting
an experiment.
Detailed record keeping is essential, to aid
in recording and reporting on the experimental
results, and supports the effectiveness and
integrity of the procedure.
They will also assist in reproducing the experimental
results, likely by others.
Traces of this approach can be seen in the
work of Hipparchus (190–120 BCE), when determining
a value for the precession of the Earth, while
controlled experiments can be seen in the
works of Jābir ibn Hayyān (721–815 CE),
al-Battani (853–929) and Alhazen (965–1039).
==== DNA-experiments ====
Watson and Crick showed an initial (and incorrect)
proposal for the structure of DNA to a team
from Kings College – Rosalind Franklin,
Maurice Wilkins, and Raymond Gosling.
Franklin immediately spotted the flaws which
concerned the water content.
Later Watson saw Franklin's detailed X-ray
diffraction images which showed an X-shape
and was able to confirm the structure was
helical.
This rekindled Watson and Crick's model building
and led to the correct structure.
..1.
DNA-characterizations
=== Evaluation and improvement ===
The scientific method is iterative.
At any stage it is possible to refine its
accuracy and precision, so that some consideration
will lead the scientist to repeat an earlier
part of the process.
Failure to develop an interesting hypothesis
may lead a scientist to re-define the subject
under consideration.
Failure of a hypothesis to produce interesting
and testable predictions may lead to reconsideration
of the hypothesis or of the definition of
the subject.
Failure of an experiment to produce interesting
results may lead a scientist to reconsider
the experimental method, the hypothesis, or
the definition of the subject.
Other scientists may start their own research
and enter the process at any stage.
They might adopt the characterization and
formulate their own hypothesis, or they might
adopt the hypothesis and deduce their own
predictions.
Often the experiment is not done by the person
who made the prediction, and the characterization
is based on experiments done by someone else.
Published results of experiments can also
serve as a hypothesis predicting their own
reproducibility.
==== DNA-iterations ====
After considerable fruitless experimentation,
being discouraged by their superior from continuing,
and numerous false starts, Watson and Crick
were able to infer the essential structure
of DNA by concrete modeling of the physical
shapes of the nucleotides which comprise it.
They were guided by the bond lengths which
had been deduced by Linus Pauling and by Rosalind
Franklin's X-ray diffraction images.
..DNA Example
=== 
Confirmation ===
Science is a social enterprise, and scientific
work tends to be accepted by the scientific
community when it has been confirmed.
Crucially, experimental and theoretical results
must be reproduced by others within the scientific
community.
Researchers have given their lives for this
vision; Georg Wilhelm Richmann was killed
by ball lightning (1753) when attempting to
replicate the 1752 kite-flying experiment
of Benjamin Franklin.To protect against bad
science and fraudulent data, government research-granting
agencies such as the National Science Foundation,
and science journals, including Nature and
Science, have a policy that researchers must
archive their data and methods so that other
researchers can test the data and methods
and build on the research that has gone before.
Scientific data archiving can be done at a
number of national archives in the U.S. or
in the World Data Center.
== Models of scientific inquiry ==
=== 
Classical model ===
The classical model of scientific inquiry
derives from Aristotle, who distinguished
the forms of approximate and exact reasoning,
set out the threefold scheme of abductive,
deductive, and inductive inference, and also
treated the compound forms such as reasoning
by analogy.
=== Hypothetico-deductive model ===
The hypothetico-deductive model or method
is a proposed description of scientific method.
Here, predictions from the hypothesis are
central: if you assume the hypothesis to be
true, what consequences follow?
If subsequent empirical investigation does
not demonstrate that these consequences or
predictions correspond to the observable world,
the hypothesis can be concluded to be false.
=== Pragmatic model ===
In 1877, Charles Sanders Peirce (1839–1914)
characterized inquiry in general not as the
pursuit of truth per se but as the struggle
to move from irritating, inhibitory doubts
born of surprises, disagreements, and the
like, and to reach a secure belief, belief
being that on which one is prepared to act.
He framed scientific inquiry as part of a
broader spectrum and as spurred, like inquiry
generally, by actual doubt, not mere verbal
or hyperbolic doubt, which he held to be fruitless.
He outlined four methods of settling opinion,
ordered from least to most successful:
The method of tenacity (policy of sticking
to initial belief) – which brings comforts
and decisiveness but leads to trying to ignore
contrary information and others' views as
if truth were intrinsically private, not public.
It goes against the social impulse and easily
falters since one may well notice when another's
opinion is as good as one's own initial opinion.
Its successes can shine but tend to be transitory.
The method of authority – which overcomes
disagreements but sometimes brutally.
Its successes can be majestic and long-lived,
but it cannot operate thoroughly enough to
suppress doubts indefinitely, especially when
people learn of other societies present and
past.
The method of the a priori – which promotes
conformity less brutally but fosters opinions
as something like tastes, arising in conversation
and comparisons of perspectives in terms of
"what is agreeable to reason."
Thereby it depends on fashion in paradigms
and goes in circles over time.
It is more intellectual and respectable but,
like the first two methods, sustains accidental
and capricious beliefs, destining some minds
to doubt it.
The scientific method – the method wherein
inquiry regards itself as fallible and purposely
tests itself and criticizes, corrects, and
improves itself.Peirce held that slow, stumbling
ratiocination can be dangerously inferior
to instinct and traditional sentiment in practical
matters, and that the scientific method is
best suited to theoretical research, which
in turn should not be trammeled by the other
methods and practical ends; reason's "first
rule" is that, in order to learn, one must
desire to learn and, as a corollary, must
not block the way of inquiry.
The scientific method excels the others by
being deliberately designed to arrive – eventually
– at the most secure beliefs, upon which
the most successful practices can be based.
Starting from the idea that people seek not
truth per se but instead to subdue irritating,
inhibitory doubt, Peirce showed how, through
the struggle, some can come to submit to truth
for the sake of belief's integrity, seek as
truth the guidance of potential practice correctly
to its given goal, and wed themselves to the
scientific method.For Peirce, rational inquiry
implies presuppositions about truth and the
real; to reason is to presuppose (and at least
to hope), as a principle of the reasoner's
self-regulation, that the real is discoverable
and independent of our vagaries of opinion.
In that vein he defined truth as the correspondence
of a sign (in particular, a proposition) to
its object and, pragmatically, not as actual
consensus of some definite, finite community
(such that to inquire would be to poll the
experts), but instead as that final opinion
which all investigators would reach sooner
or later but still inevitably, if they were
to push investigation far enough, even when
they start from different points.
In tandem he defined the real as a true sign's
object (be that object a possibility or quality,
or an actuality or brute fact, or a necessity
or norm or law), which is what it is independently
of any finite community's opinion and, pragmatically,
depends only on the final opinion destined
in a sufficient investigation.
That is a destination as far, or near, as
the truth itself to you or me or the given
finite community.
Thus, his theory of inquiry boils down to
"Do the science."
Those conceptions of truth and the real involve
the idea of a community both without definite
limits (and thus potentially self-correcting
as far as needed) and capable of definite
increase of knowledge.
As inference, "logic is rooted in the social
principle" since it depends on a standpoint
that is, in a sense, unlimited.Paying special
attention to the generation of explanations,
Peirce outlined the scientific method as a
coordination of three kinds of inference in
a purposeful cycle aimed at settling doubts,
as follows (in §III–IV in "A Neglected
Argument" except as otherwise noted):
Abduction (or retroduction).
Guessing, inference to explanatory hypotheses
for selection of those best worth trying.
From abduction, Peirce distinguishes induction
as inferring, on the basis of tests, the proportion
of truth in the hypothesis.
Every inquiry, whether into ideas, brute facts,
or norms and laws, arises from surprising
observations in one or more of those realms
(and for example at any stage of an inquiry
already underway).
All explanatory content of theories comes
from abduction, which guesses a new or outside
idea so as to account in a simple, economical
way for a surprising or complicative phenomenon.
Oftenest, even a well-prepared mind guesses
wrong.
But the modicum of success of our guesses
far exceeds that of sheer luck and seems born
of attunement to nature by instincts developed
or inherent, especially insofar as best guesses
are optimally plausible and simple in the
sense, said Peirce, of the "facile and natural",
as by Galileo's natural light of reason and
as distinct from "logical simplicity".
Abduction is the most fertile but least secure
mode of inference.
Its general rationale is inductive: it succeeds
often enough and, without it, there is no
hope of sufficiently expediting inquiry (often
multi-generational) toward new truths.
Coordinative method leads from abducing a
plausible hypothesis to judging it for its
testability and for how its trial would economize
inquiry itself.
Peirce calls his pragmatism "the logic of
abduction".
His pragmatic maxim is: "Consider what effects
that might conceivably have practical bearings
you conceive the objects of your conception
to have.
Then, your conception of those effects is
the whole of your conception of the object".
His pragmatism is a method of reducing conceptual
confusions fruitfully by equating the meaning
of any conception with the conceivable practical
implications of its object's conceived effects
– a method of experimentational mental reflection
hospitable to forming hypotheses and conducive
to testing them.
It favors efficiency.
The hypothesis, being insecure, needs to have
practical implications leading at least to
mental tests and, in science, lending themselves
to scientific tests.
A simple but unlikely guess, if uncostly to
test for falsity, may belong first in line
for testing.
A guess is intrinsically worth testing if
it has instinctive plausibility or reasoned
objective probability, while subjective likelihood,
though reasoned, can be misleadingly seductive.
Guesses can be chosen for trial strategically,
for their caution (for which Peirce gave as
example the game of Twenty Questions), breadth,
and incomplexity.
One can hope to discover only that which time
would reveal through a learner's sufficient
experience anyway, so the point is to expedite
it; the economy of research is what demands
the leap, so to speak, of abduction and governs
its art.
Deduction.
Two stages:
Explication.
Unclearly premissed, but deductive, analysis
of the hypothesis in order to render its parts
as clear as possible.
Demonstration: Deductive Argumentation, Euclidean
in procedure.
Explicit deduction of hypothesis's consequences
as predictions, for induction to test, about
evidence to be found.
Corollarial or, if needed, theorematic.
Induction.
The long-run validity of the rule of induction
is deducible from the principle (presuppositional
to reasoning in general) that the real is
only the object of the final opinion to which
adequate investigation would lead; anything
to which no such process would ever lead would
not be real.
Induction involving ongoing tests or observations
follows a method which, sufficiently persisted
in, will diminish its error below any predesignate
degree.
Three stages:
Classification.
Unclearly premissed, but inductive, classing
of objects of experience under general ideas.
Probation: direct inductive argumentation.
Crude (the enumeration of instances) or gradual
(new estimate of proportion of truth in the
hypothesis after each test).
Gradual induction is qualitative or quantitative;
if qualitative, then dependent on weightings
of qualities or characters; if quantitative,
then dependent on measurements, or on statistics,
or on countings.
Sentential Induction.
"...which, by inductive reasonings, appraises
the different probations singly, then their
combinations, then makes self-appraisal of
these very appraisals themselves, and passes
final judgment on the whole result".
== Science of complex systems ==
Science applied to complex systems can involve
elements such as transdisciplinarity, systems
theory and scientific modelling.
The Santa Fe Institute studies such systems;
Murray Gell-Mann interconnects these topics
with message passing.In general, the scientific
method may be difficult to apply stringently
to diverse, interconnected systems and large
data sets.
In particular, practices used within Big data,
such as predictive analytics, may be considered
to be at odds with the scientific method.
== Communication and community ==
Frequently the scientific method is employed
not only by a single person, but also by several
people cooperating directly or indirectly.
Such cooperation can be regarded as an important
element of a scientific community.
Various standards of scientific methodology
are used within such an environment.
=== Peer review evaluation ===
Scientific journals use a process of peer
review, in which scientists' manuscripts are
submitted by editors of scientific journals
to (usually one to three, and usually anonymous)
fellow scientists familiar with the field
for evaluation.
In certain journals, the journal itself selects
the referees; while in others (especially
journals that are extremely specialized),
the manuscript author might recommend referees.
The referees may or may not recommend publication,
or they might recommend publication with suggested
modifications, or sometimes, publication in
another journal.
This standard is practiced to various degrees
by different journals, and can have the effect
of keeping the literature free of obvious
errors and to generally improve the quality
of the material, especially in the journals
who use the standard most rigorously.
The peer review process can have limitations
when considering research outside the conventional
scientific paradigm: problems of "groupthink"
can interfere with open and fair deliberation
of some new research.
=== Documentation and replication ===
Sometimes experimenters may make systematic
errors during their experiments, veer from
standard methods and practices (Pathological
science) for various reasons, or, in rare
cases, deliberately report false results.
Occasionally because of this then, other scientists
might attempt to repeat the experiments in
order to duplicate the results.
==== Archiving ====
Researchers sometimes practice scientific
data archiving, such as in compliance with
the policies of government funding agencies
and scientific journals.
In these cases, detailed records of their
experimental procedures, raw data, statistical
analyses and source code can be preserved
in order to provide evidence of the methodology
and practice of the procedure and assist in
any potential future attempts to reproduce
the result.
These procedural records may also assist in
the conception of new experiments to test
the hypothesis, and may prove useful to engineers
who might examine the potential practical
applications of a discovery.
==== Data sharing ====
When additional information is needed before
a study can be reproduced, the author of the
study might be asked to provide it.
They might provide it, or if the author refuses
to share data, appeals can be made to the
journal editors who published the study or
to the institution which funded the research.
==== Limitations ====
Since it is impossible for a scientist to
record everything that took place in an experiment,
facts selected for their apparent relevance
are reported.
This may lead, unavoidably, to problems later
if some supposedly irrelevant feature is questioned.
For example, Heinrich Hertz did not report
the size of the room used to test Maxwell's
equations, which later turned out to account
for a small deviation in the results.
The problem is that parts of the theory itself
need to be assumed in order to select and
report the experimental conditions.
The observations are hence sometimes described
as being 'theory-laden'.
=== Dimensions of practice ===
The primary constraints on contemporary science
are:
Publication, i.e.
Peer review
Resources (mostly funding)It has not always
been like this: in the old days of the "gentleman
scientist" funding (and to a lesser extent
publication) were far weaker constraints.
Both of these constraints indirectly require
scientific method – work that violates the
constraints will be difficult to publish and
difficult to get funded.
Journals require submitted papers to conform
to "good scientific practice" and to a degree
this can be enforced by peer review.
Originality, importance and interest are more
important – see for example the author guidelines
for Nature.
Smaldino and McElreath 2016 have noted that
our need to reward scientific understanding
is being nullified by poor research design
and poor data analysis, which is leading to
false-positive findings.
== Philosophy and sociology of science ==
=== 
Analytical philosophy ===
Philosophy of science looks at the underpinning
logic of the scientific method, at what separates
science from non-science, and the ethic that
is implicit in science.
There are basic assumptions, derived from
philosophy by at least one prominent scientist,
that form the base of the scientific method
– namely, that reality is objective and
consistent, that humans have the capacity
to perceive reality accurately, and that rational
explanations exist for elements of the real
world.
These assumptions from methodological naturalism
form a basis on which science may be grounded.
Logical Positivist, empiricist, falsificationist,
and other theories have criticized these assumptions
and given alternative accounts of the logic
of science, but each has also itself been
criticized.
Thomas Kuhn examined the history of science
in his The Structure of Scientific Revolutions,
and found that the actual method used by scientists
differed dramatically from the then-espoused
method.
His observations of science practice are essentially
sociological and do not speak to how science
is or can be practiced in other times and
other cultures.
Norwood Russell Hanson, Imre Lakatos and Thomas
Kuhn have done extensive work on the "theory
laden" character of observation.
Hanson (1958) first coined the term for the
idea that all observation is dependent on
the conceptual framework of the observer,
using the concept of gestalt to show how preconceptions
can affect both observation and description.
He opens Chapter 1 with a discussion of the
Golgi bodies and their initial rejection as
an artefact of staining technique, and a discussion
of Brahe and Kepler observing the dawn and
seeing a "different" sun rise despite the
same physiological phenomenon.
Kuhn and Feyerabend acknowledge the pioneering
significance of his work.
Kuhn (1961) said the scientist generally has
a theory in mind before designing and undertaking
experiments so as to make empirical observations,
and that the "route from theory to measurement
can almost never be traveled backward".
This implies that the way in which theory
is tested is dictated by the nature of the
theory itself, which led Kuhn (1961, p. 166)
to argue that "once it has been adopted by
a profession ... no theory is recognized to
be testable by any quantitative tests that
it has not already passed".
=== Post-modernism and science wars ===
Paul Feyerabend similarly examined the history
of science, and was led to deny that science
is genuinely a methodological process.
In his book Against Method he argues that
scientific progress is not the result of applying
any particular method.
In essence, he says that for any specific
method or norm of science, one can find a
historic episode where violating it has contributed
to the progress of science.
Thus, if believers in scientific method wish
to express a single universally valid rule,
Feyerabend jokingly suggests, it should be
'anything goes'.
Criticisms such as his led to the strong programme,
a radical approach to the sociology of science.
The postmodernist critiques of science have
themselves been the subject of intense controversy.
This ongoing debate, known as the science
wars, is the result of conflicting values
and assumptions between the postmodernist
and realist camps.
Whereas postmodernists assert that scientific
knowledge is simply another discourse (note
that this term has special meaning in this
context) and not representative of any form
of fundamental truth, realists in the scientific
community maintain that scientific knowledge
does reveal real and fundamental truths about
reality.
Many books have been written by scientists
which take on this problem and challenge the
assertions of the postmodernists while defending
science as a legitimate method of deriving
truth.
=== Anthropology ===
In anthropology, following the anthropological
fieldworks in an academic scientific laboratory
by Latour and Woolgar, Karin Knorr Cetina
has conducted a comparative anthropological
study of two scientific fields (namely high
energy physics and molecular biology) to conclude
that the epistemic practices and reasonings
within both scientific communities are different
enough to introduce the concept of "epistemic
cultures", in contradiction with the idea
that a so-called "scientific method" is unique
and a unifying concept.
=== Role of chance in discovery ===
Somewhere between 33% and 50% of all scientific
discoveries are estimated to have been stumbled
upon, rather than sought out.
This may explain why scientists so often express
that they were lucky.
Louis Pasteur is credited with the famous
saying that "Luck favours the prepared mind",
but some psychologists have begun to study
what it means to be 'prepared for luck' in
the scientific context.
Research is showing that scientists are taught
various heuristics that tend to harness chance
and the unexpected.
This is what Nassim Nicholas Taleb calls "Anti-fragility";
while some systems of investigation are fragile
in the face of human error, human bias, and
randomness, the scientific method is more
than resistant or tough – it actually benefits
from such randomness in many ways (it is anti-fragile).
Taleb believes that the more anti-fragile
the system, the more it will flourish in the
real world.Psychologist Kevin Dunbar says
the process of discovery often starts with
researchers finding bugs in their experiments.
These unexpected results lead researchers
to try to fix what they think is an error
in their method.
Eventually, the researcher decides the error
is too persistent and systematic to be a coincidence.
The highly controlled, cautious and curious
aspects of the scientific method are thus
what make it well suited for identifying such
persistent systematic errors.
At this point, the researcher will begin to
think of theoretical explanations for the
error, often seeking the help of colleagues
across different domains of expertise.
== Relationship with mathematics ==
Science is the process of gathering, comparing,
and evaluating proposed models against observables.
A model can be a simulation, mathematical
or chemical formula, or set of proposed steps.
Science is like mathematics in that researchers
in both disciplines try to distinguish what
is known from what is unknown at each stage
of discovery.
Models, in both science and mathematics, need
to be internally consistent and also ought
to be falsifiable (capable of disproof).
In mathematics, a statement need not yet be
proven; at such a stage, that statement would
be called a conjecture.
But when a statement has attained mathematical
proof, that statement gains a kind of immortality
which is highly prized by mathematicians,
and for which some mathematicians devote their
lives.Mathematical work and scientific work
can inspire each other.
For example, the technical concept of time
arose in science, and timelessness was a hallmark
of a mathematical topic.
But today, the Poincaré conjecture has been
proven using time as a mathematical concept
in which objects can flow (see Ricci flow).
Nevertheless, the connection between mathematics
and reality (and so science to the extent
it describes reality) remains obscure.
Eugene Wigner's paper, The Unreasonable Effectiveness
of Mathematics in the Natural Sciences, is
a very well known account of the issue from
a Nobel Prize-winning physicist.
In fact, some observers (including some well
known mathematicians such as Gregory Chaitin,
and others such as Lakoff and Núñez) have
suggested that mathematics is the result of
practitioner bias and human limitation (including
cultural ones), somewhat like the post-modernist
view of science.
George Pólya's work on problem solving, the
construction of mathematical proofs, and heuristic
show that the mathematical method and the
scientific method differ in detail, while
nevertheless resembling each other in using
iterative or recursive steps.
In Pólya's view, understanding involves restating
unfamiliar definitions in your own words,
resorting to geometrical figures, and questioning
what we know and do not know already; analysis,
which Pólya takes from Pappus, involves free
and heuristic construction of plausible arguments,
working backward from the goal, and devising
a plan for constructing the proof; synthesis
is the strict Euclidean exposition of step-by-step
details of the proof; review involves reconsidering
and re-examining the result and the path taken
to it.
Gauss, when asked how he came about his theorems,
once replied "durch planmässiges Tattonieren"
(through systematic palpable experimentation).Imre
Lakatos argued that mathematicians actually
use contradiction, criticism and revision
as principles for improving their work.
In like manner to science, where truth is
sought, but certainty is not found, in Proofs
and refutations (1976), what Lakatos tried
to establish was that no theorem of informal
mathematics is final or perfect.
This means that we should not think that a
theorem is ultimately true, only that no counterexample
has yet been found.
Once a counterexample, i.e. an entity contradicting/not
explained by the theorem is found, we adjust
the theorem, possibly extending the domain
of its validity.
This is a continuous way our knowledge accumulates,
through the logic and process of proofs and
refutations.
(If axioms are given for a branch of mathematics,
however, Lakatos claimed that proofs from
those axioms were tautological, i.e. logically
true, by rewriting them, as did Poincaré
(Proofs and Refutations, 1976).)
Lakatos proposed an account of mathematical
knowledge based on Polya's idea of heuristics.
In Proofs and Refutations, Lakatos gave several
basic rules for finding proofs and counterexamples
to conjectures.
He thought that mathematical 'thought experiments'
are a valid way to discover mathematical conjectures
and proofs.
=== Relationship with statistics ===
The scientific method has been extremely successful
in bringing the world out of medieval thinking,
especially once it was combined with industrial
processes.
However, when the scientific method employs
statistics as part of its arsenal, there are
mathematical and practical issues that can
have a deleterious effect on the reliability
of the output of scientific methods.
This is described in a popular 2005 scientific
paper "Why Most Published Research Findings
Are False" by John Ioannidis.The particular
points raised are statistical ("The smaller
the studies conducted in a scientific field,
the less likely the research findings are
to be true" and "The greater the flexibility
in designs, definitions, outcomes, and analytical
modes in a scientific field, the less likely
the research findings are to be true.") and
economical ("The greater the financial and
other interests and prejudices in a scientific
field, the less likely the research findings
are to be true" and "The hotter a scientific
field (with more scientific teams involved),
the less likely the research findings are
to be true.")
Hence: "Most research findings are false for
most research designs and for most fields"
and "As shown, the majority of modern biomedical
research is operating in areas with very low
pre- and poststudy probability for true findings."
However: "Nevertheless, most new discoveries
will continue to stem from hypothesis-generating
research with low or very low pre-study odds,"
which means that *new* discoveries will come
from research that, when that research started,
had low or very low odds (a low or very low
chance) of succeeding.
Hence, if the scientific method is used to
expand the frontiers of knowledge, research
into areas that are outside the mainstream
will yield most new discoveries.
== See also ==
=== 
Problems and issues ===
=== History, philosophy, sociology ===
== 
Notes
