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Scientific method

Scientific method refers to a body of techniques for investigating phenomena, acquiring new[1] knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of reasoning.[2] A scientific method consists of the collection of data through observation and experimentation, and the formulation and testing of hypotheses.[3]

Although procedures vary from one field of inquiry to another, identifiable features distinguish scientific inquiry from other methodologies of knowledge. Scientific researchers propose hypotheses as explanations of phenomena, and design experimental studies to test these hypotheses. These steps must be repeatable in order to dependably predict any future results. Theories that encompass wider domains of inquiry may bind many independently derived hypotheses together in a coherent, supportive structure. This in turn may help form new hypotheses or place groups of hypotheses into context.

Among other facets shared by the various fields of inquiry is the conviction that the process must be objective to reduce biased interpretations of the results. Another basic expectation is to document, archive and share all data and methodology so they are available for careful scrutiny by other scientists, thereby allowing other researchers the opportunity to verify results by attempting to reproduce them. This practice, called full disclosure, also allows statistical measures of the reliability of these data to be established.

[edit] Introduction to scientific method

Ibn al-Haytham (Alhazen), 965â1039, Basra.
"Modern science owes its origins and present flourishing state to a new scientific method which was fashioned almost entirely by Galileo Galilei (1564-1642)" âMorris Kline[4]
Johannes Kepler (1571â1630). "Kepler shows his keen logical sense in detailing the whole process by which he finally arrived at the true orbit. This is the greatest piece of Retroductive reasoning ever performed." âC. S. Peirce, circa 1896, on Kepler's reasoning through explanatory hypotheses[5]

Since Ibn al-Haytham (Alhazen, 965â1039), one of the key figures in the development of scientific method, the emphasis has been on seeking truth:

Truth is sought for its own sake. And those who are engaged upon the quest for anything for its own sake are not interested in other things. Finding the truth is difficult, and the road to it is rough.[6]
"Light travels through transparent bodies in straight lines only" â Alhazen in Book of Optics (1021).
How does light travel through transparent bodies? Light travels through transparent bodies in straight lines only.... We have explained this exhaustively in our Book of Optics. But let us now mention something to prove this convincingly: the fact that light travels in straight lines is clearly observed in the lights which enter into dark rooms through holes.... [T]he entering light will be clearly observable in the dust which fills the air.[7]

The conjecture that "light travels through transparent bodies in straight lines only" was corroborated by Alhazen only after years of effort. His demonstration of the conjecture was to place a straight stick or a taut thread next to the light beam,[8] to prove that light travels in a straight line.

Scientific methodology has been practiced in some form for at least one thousand years. There are difficulties in a formulaic statement of method, however. As William Whewell (1794â1866) noted in his History of Inductive Science (1837) and in Philosophy of Inductive Science (1840), "invention, sagacity, genius" are required at every step in scientific method. It is not enough to base scientific method on experience alone;[9] multiple steps are needed in scientific method, ranging from our experience to our imagination, back and forth.

In the twentieth century, a hypothetico-deductive model for scientific method was formulated (for a more formal discussion, see below):

1. Use your experience: Consider the problem and try to make sense of it. Look for previous explanations. If this is a new problem to you, then move to step 2.
2. Form a conjecture: When nothing else is yet known, try to state an explanation, to someone else, or to your notebook.
3. Deduce a prediction from that explanation: If you assume 2 is true, what consequences follow?
4. Test: Look for the opposite of each consequence in order to disprove 2. It is a logical error to seek 3 directly as proof of 2. This error is called affirming the consequent.[10]

This model underlies the scientific revolution. One thousand years ago, Alhazen demonstrated the importance of steps 1 and 4. Galileo (1638) also showed the importance of step 4 (also called Experiment) in Two New Sciences. One possible sequence in this model would be 1, 2, 3, 4. If the outcome of 4 holds, and 3 is not yet disproven, you may continue with 3, 4, 1, and so forth; but if the outcome of 4 shows 3 to be false, you will have go back to 2 and try to invent a new 2, deduce a new 3, look for 4, and so forth.

Note that this method can never absolutely verify (prove the truth of) 2. It can only falsify 2.[11] (This is what Einstein meant when he said "No amount of experimentation can ever prove me right; a single experiment can prove me wrong."[12]) However, as pointed out by Carl Hempel (1905â1997) this simple view of scientific method is incomplete; the formulation of the conjecture might itself be the result of inductive reasoning. Thus the likelihood of the prior observation being true is statistical in nature [13] and would strictly require a Bayesian analysis. To overcome this uncertainty, experimental scientists must formulate a crucial experiment, in order for it to corroborate a more likely hypothesis.

In the twentieth century, Ludwik Fleck (1896â1961) and others found that we need to consider our experiences more carefully, because our experience may be biased, and that we need to be more exact when describing our experiences.[14] These considerations are discussed below.

[edit] DNA example

The Keystones of Science project, sponsored by the journal Science, has selected a number of scientific articles from that journal and annotated them, illustrating how different parts of each article embody scientific method. Here [15] is an annotated example of this scientific method example titled "Microbial Genes in the Human Genome: Lateral Transfer or Gene Loss?".

DNA icon (25x25).png Four basic elements of scientific method are illustrated below, by example from the discovery of the structure of DNA:

The examples are continued in "Evaluations and iterations" with DNA-iterations.[20]

[edit] Truth and belief

Flying gallop falsified; see image below.

Belief can alter observations; those with a particular belief will often see things as reinforcing their belief, even if to another observer they would appear not to do so. Even researchers admit that the first observation may have been a little imprecise, whereas the second and third were "adjusted to the facts," until tradition, education, and familiarity produce a readiness for new perception.[21]

Needham's Science and Civilization in China uses the 'flying gallop' image as an example of observation:[22] in it, the legs of a galloping horse are depicted as splayed, when the stop-action pictures of a horse's gallop by Eadweard Muybridge show otherwise. In a gallop, at the moment that no hoof is touching the ground, a horse's legs are gathered together and are not splayed. Earlier paintings depict the incorrect flying gallop observation (this is an example of observer bias).

This demonstrates Ludwik Fleck's caution that people observe what they expect to observe, until shown otherwise; our beliefs will affect our observations (and therefore our subsequent actions). The purpose of a scientific method is to test a hypothesis, a proposed explanation about how things are, via repeatable experimental observations which can definitively contradict the hypothesis.

[edit] Elements of 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.

Four essential elements[23][24][25] of a scientific method[26] are iterations,[27][28] recursions,[29] interleavings, and orderings of the following:

Each element of a 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). The elements above are often taught in the educational system.[36]

Scientific method is not a recipe: it requires intelligence, imagination, and creativity.[37] It is also 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 reduced out from Einstein's theories â all phenomena that Newton could not have observed â Newton's equations remain. Einstein's theories are expansions and refinements of Newton's theories and, thus, increase our confidence in Newton's work.

A linearized, pragmatic scheme of the four points above is sometimes offered as a guideline for proceeding:[38]

  1. Define the question
  2. Gather information and resources (observe)
  3. Form hypothesis
  4. Perform experiment and collect data
  5. Analyze data
  6. Interpret data and draw conclusions that serve as a starting point for new hypothesis
  7. Publish results
  8. Retest (frequently done by other scientists)

The iterative cycle inherent in this step-by-step methodology goes from point 3 to 6 back to 3 again.

While this schema outlines a typical hypothesis/testing method,[39] it should also be noted that a number of philosophers, historians and sociologists of science (perhaps most notably Paul Feyerabend) claim that such descriptions of scientific method have little relation to the ways science is actually practiced.

The "operational" paradigm combines the concepts of operational definition, instrumentalism, and utility:

The essential elements of a scientific method are operations, observations, models, and a utility function for evaluating models.[40][not in citation given]

[edit] Characterizations

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 correctly characterized St. Elmo's fire as electrical in nature, but it has taken a long series of experiments and theory to establish this. While seeking the pertinent properties of the subjects, this 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 a science, such as chemistry or biology. Scientific measurements taken 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, or voltmeters, and the progress of a scientific field is usually intimately tied to their invention and development.

"I am not accustomed to saying anything with certainty after only one or two observations."âAndreas Vesalius (1546) [41]

[edit] 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 that are used. Counts of things, such as the number of people in a nation at a particular time, may also have an uncertainty due to limitations of the method used. Counts may only represent a sample of desired quantities, with an uncertainty that depends upon the sampling method used and the number of samples taken.

[edit] 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, electrical 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 certain 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 sometimes arise upon realizing that certain terms had 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.[42] 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.

[edit] Example of characterizations

[edit] DNA-characterizations
DNA icon (25x25).png

The history of the discovery of the structure of DNA is a classic example of the elements of scientific method: in 1950 it was known that genetic inheritance had a mathematical description, starting with the studies of Gregor Mendel. But the mechanism of the gene 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 which were painstakingly assembled over the course of 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

[edit] Precession of Mercury
Precession of the perihelion (exaggerated)

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 record the motion of planet Earth. Newton was able to condense these measurements into consequences of his laws of motion. But the perihelion of the planet Mercury's orbit exhibits a precession that is not fully explained by Newton's laws of motion. The observed difference for Mercury's precession between Newtonian theory and relativistic theory (approximately 43 arc-seconds per century), was one of the things that occurred to Einstein as a possible early test of his theory of General Relativity.

[edit] 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 â their own creativity, ideas from other fields, induction, Bayesian inference, and so on â to imagine possible explanations for a phenomenon under study. 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 â bald suppositions and areas of vagueness.[43]

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 making these determinations.

[edit] DNA-hypotheses
DNA icon (25x25).png

Linus Pauling proposed that DNA might be a triple helix.[44] 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[45] 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 raceâsee section on "DNA-predictions" below)

[edit] 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 only talk about probabilities.

It is essential that the outcome be currently unknown. Only in this case does the eventuation increase the probability that the hypothesis be true. If the outcome is already known, it's 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 useful for the method, and must wait for others who might come afterward, and perhaps rekindle its line of reasoning. For example, a new technology or theory might make the necessary experiments feasible.

[edit] DNA-predictions
DNA icon (25x25).png

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'.[46][47] This prediction followed from the work of Cochran, Crick and Vand[18] (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.

Also in their first paper, Watson and Crick predicted that the double helix structure 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

[edit] General relativity

Einstein's theory of General Relativity makes several specific predictions about the observable structure of space-time, such as a prediction 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.[48]

[edit] Experiments

Once predictions are made, they can be tested by experiments. If test results contradict predictions, then the hypotheses are called into question and explanations may be sought. Sometimes experiments are conducted incorrectly and are at fault. If the results confirm the predictions, then the hypotheses are considered likely to be correct but might still be wrong and are 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 providing evidence of the effectiveness and integrity of the procedure. They will also assist in reproducing the experimental results. Traces of this tradition 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 Muslim scientists such as Jäbir ibn Hayyän (721-815 CE), al-Battani (853â929) and Alhacen (965-1039).

[edit] DNA-experiments
DNA icon (25x25).png

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 confirmed that the structure was helical.[19][49] This rekindled Watson and Crick's model building and led to the correct structure. ..1. DNA-characterizations

[edit] Evaluation and improvement

The scientific process is iterative. At any stage it is possible 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 they are considering. 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 the experiment to produce interesting results may lead the scientist to reconsidering 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.

[edit] DNA-iterations
DNA icon (25x25).png

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.[20][50] 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

[edit] Confirmation

Science is a social enterprise, and scientific work tends to be accepted by the community when it has been confirmed. Crucially, experimental and theoretical results must be reproduced by others within the science 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.[51]

To protect against bad science and fraudulent data, government research granting agencies like NSF and science journals like Nature and Science have a policy that researchers must archive their data and methods so other researchers can access it, 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.

[edit] Models of scientific inquiry

[edit] Classical model

The classical model of scientific inquiry derives from Aristotle,[52] 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.

[edit] Pragmatic model

In 1877[53], Charles Sanders Peirce (pronounced /ˈpÉrs/ purse) (1839â1914) 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 doubt (such as hyperbolic doubt), which he held to be fruitless. He outlined four methods of settling opinion, ordered from least to most successful:

  1. The method of tenacity (sticking with one's initial belief through stubbornness) â which leads to trying to ignore contrary information and others' views as if truth were intrinsically private, not public.
  2. The method of authority â which overcomes disagreements but sometimes brutally.
  3. The method of congruity or the a priori or the dilettante or "what is agreeable to reason" â which promotes conformity less brutally but depends on taste and fashion in paradigms, fosters barren disputation and, like the first two methods, fails to advance knowledge.
  4. The scientific method â the method wherein inquiry regards itself as fallible and actually tests itself and criticizes, corrects, and improves itself.

Peirce held that slow, stumbling ratiocination can be dangerously inferior to instinct, tradition, and sentiment in practical matters, and that the scientific method is best suited to theoretical research,[54] 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.[55] The scientific method excels the others by being deliberately designed to arrive â eventually â 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 such doubt can lead some to submit to truth and seek truth as simply that opinion which would lead consequent practice to its goal and not astray.[53][56]

In Peirce's view, the conception of rational inquiry depends on, but also informs, the conceptions of 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 truth is discoverable and independent of our vagaries of opinion. He defined truth as the correspondence of a sign (in particular, a proposition) to its object and, pragmatically, not as any actual consensus of any definite, finite community (such that to inquire would be to poll the experts), but instead as that ideal final opinion which all reasonable scientific intelligences would reach sooner or later but still inevitably, if they were to push investigation far enough, even when they start from different points.[57] 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 ideal final opinion. That ideal consensus or opinion is as far, or near, as the truth itself to you or me or any finite community. Thus his theory of inquiry boils down to "Do the science." At the same time those conceptions of truth and the real involve the idea of a community, both without definite limits and capable of definite increase of knowledge.[58] As inference, "logic is rooted in the social principle" since it depends on a standpoint that is, in a sense, unlimited.[59]

Paying special attention to the generation of explanations, Peirce outlined scientific method as a collaboration of three kinds of inference in a purposeful cycle aimed at settling doubts, as follows:[60]

1. Abduction (or retroduction). Guessing, inference to the explanatory hypothesis 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) and the pondering all the phenomenon's aspects in the attempt to resolve the wonder. All explanatory content of theories is reached through abduction, the most insecure among modes of inference. 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, for which the economics of research demands and even governs the "leap" of abduction,[61] whose modicum of success depends on one's being somehow attuned to nature by instincts developed and likely inborn. Abduction has an overall inductive rationale in that it works often enough and that nothing else works,[62] at least not quickly enough when science is already properly rather slow, the work of indefinitely many generations. Peirce calls his pragmatism "the logic of abduction".[63] 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".[57] His pragmatism is a method of reducing conceptual confusions by equating the meaning of any conception with the conceivable practical implications of its object's conceived effects. It is a method of experimentational mental reflection arriving at conceptions in terms of conceivable confirmatory and disconfirmatory circumstances â a method hospitable to the generation of explanatory hypotheses, and conducive to the employment and improvement of verification to test the truth of putative knowledge. Given abduction's dependence on instinctive attunement to nature and its aim to economize inquiry, its explanatory hypotheses should be plausible and indeed optimally simple in the sense of "facile and natural" (for which Peirce cites Galileo and which Peirce distinguishes from "logical simplicity"). Given abduction's insecurity, it should imply consequences with conceivable practical bearing leading at least to mental tests, and, in science, lending themselves to scientific testing.

2. Deduction. Analysis of hypothesis and deduction of distinctive consequences (for induction to test so as to evaluate the hypothesis). Two stages:

i. Explication. Logical analysis of the hypothesis in order to render its parts as clear as possible.
ii. Demonstration (or deductive argumentation). Deduction of hypothesis's consequence. Corollarial or, if needed, Theorematic.

3. Induction. The long-run validity of the rule of induction is deducible from the principle (presuppositional to reasoning in general[57]) that the real is only the object of the final opinion to which adequate investigation would lead[64] Induction involving ongoing tests or observations follows a method which, sufficiently persisted in, will diminish its error below any predesignate degree[60] and, if there were something to which such a process would never lead, then that thing would not be real. Three stages:

i. Classification. Classing objects of experience under general ideas.
ii. Probation (or 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 Quantitative, then dependent on measurements, or on statistics, or on countings.
iii. 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".[60]

[edit] Computational approaches

Many subspecialties of applied logic and computer science, such as artificial intelligence, machine learning, computational learning theory, inferential statistics, and knowledge representation, are concerned with setting out computational, logical, and statistical frameworks for the various types of inference involved in scientific inquiry. In particular, they contribute hypothesis formation, logical deduction, and empirical testing. Some of these applications draw on measures of complexity from algorithmic information theory to guide the making of predictions from prior distributions of experience, for example, see the complexity measure called the speed prior from which a computable strategy for optimal inductive reasoning can be derived.

[edit] Communication, community, culture

Frequently a 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 one of the defining elements of a scientific community. Various techniques have been developed to ensure the integrity of that scientific method within such an environment.

[edit] 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) fellow (usually anonymous) scientists familiar with the field for evaluation. The referees may or may not recommend publication, publication with suggested modifications, or, sometimes, publication in another journal. This serves to keep the scientific literature free of unscientific or crackpot work, helps to cut down on obvious errors, and generally otherwise improve the quality of the scientific literature.

[edit] Documentation and replication

Sometimes experimenters may make systematic errors during their experiments, unconsciously veer from a scientific method (Pathological science) for various reasons, or, in rare cases, deliberately falsify their results. Consequently, it is a common practice for other scientists to attempt to repeat the experiments in order to duplicate the results, thus further validating the hypothesis.

[edit] Archiving

As a result, researchers are expected to practice scientific data archiving in compliance with the policies of government funding agencies and scientific journals. Detailed records of their experimental procedures, raw data, statistical analyses and source code are preserved in order to provide evidence of the effectiveness and integrity of the procedure and assist in reproduction. 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.

[edit] Data sharing

When additional information is needed before a study can be reproduced, the author of the study is expected to provide it promptly - although a small charge may apply. 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.

[edit] 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'.

[edit] Dimensions of practice

The primary constraints on contemporary western science are:

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 bring in a scientific method â work that too obviously violates the constraints will be difficult to publish and difficult to get funded. Journals do not require submitted papers to conform to anything more specific than "good scientific practice" and this is mostly enforced by peer review. Originality, importance and interest are more important - see for example the author guidelines for Nature.

Criticisms (see Critical theory) of these restraints are that they are so nebulous in definition (e.g. "good scientific practice") and open to ideological, or even political, manipulation apart from a rigorous practice of a scientific method, that they often serve to censor rather than promote scientific discovery.[citation needed] Apparent censorship through refusal to publish ideas unpopular with mainstream scientists (unpopular because of ideological reasons and/or because they seem to contradict long held scientific theories) has soured the popular perception of scientists as being neutral or seekers of truth and often denigrated popular perception of science as a whole.

[edit] Philosophy and sociology of science

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 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 the basis on which science is grounded. Logical Positivist, empiricist, falsificationist, and other theories have claimed to give a definitive account of the logic of science, but each has in turn been criticized.

Thomas Samuel 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.

Imre Lakatos and Thomas Kuhn have done extensive work on the "theory laden" character of observation. 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".

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 "anything goes", by which he meant that for any specific methodology or norm of science, successful science has been done in violation of it. Criticisms such as his led to the strong programme, a radical approach to the sociology of science.

In his 1958 book, Personal Knowledge, chemist and philosopher Michael Polanyi (1891â1976) criticized the common view that the scientific method is purely objective and generates objective knowledge. Polanyi cast this view as a misunderstanding of the scientific method and of the nature of scientific inquiry, generally. He argued that scientists do and must follow personal passions in appraising facts and in determining which scientific questions to investigate. He concluded that a structure of liberty is essential for the advancement of science - that the freedom to pursue science for its own sake is a prerequisite for the production of knowledge through peer review and the scientific method.

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.[65]

[edit] History

The development of the scientific method is inseparable from the history of science itself. Ancient Egyptian documents describe empirical methods in astronomy,[66] mathematics,[67] and medicine.[68] The ancient Greek philosopher Thales in the 6th century BC refused to accept supernatural, religious or mythological explanations for natural phenomena, proclaiming that every event had a natural cause. The development of deductive reasoning by Plato was an important step towards the scientific method. Empiricism seems to have been formalized by Aristotle, who believed that universal truths could be reached via induction.

There are hints of experimental methods from the Classical world (e.g., those reported by Archimedes in a report recovered early in the 20th century CE from an overwritten manuscript), but the first clear instances of an experimental scientific method seem to have been developed in the Arabic world, by Muslim scientists, who introduced the use of experimentation and quantification to distinguish between competing scientific theories set within a generally empirical orientation, perhaps by Alhazen in his optical experiments reported in his Book of Optics (1021).[69] The modern scientific method crystallized no later than in the 17th and 18th centuries. In his work Novum Organum (1620) â a reference to Aristotle's Organon â Francis Bacon outlined a new system of logic to improve upon the old philosophical process of syllogism. Then, in 1637, René Descartes established the framework for a scientific method's guiding principles in his treatise, Discourse on Method. The writings of Alhazen, Bacon and Descartes are considered critical in the historical development of the modern scientific method, as are those of John Stuart Mill.[70]

In the late 19th century, Charles Sanders Peirce proposed a schema that would turn out to have considerable influence in the development of current scientific method generally. Peirce accelerated the progress on several fronts. Firstly, speaking in broader context in "How to Make Our Ideas Clear" (1878), Peirce outlined an objectively verifiable method to test the truth of putative knowledge on a way that goes beyond mere foundational alternatives, focusing upon both deduction and induction. He thus placed induction and deduction in a complementary rather than competitive context (the latter of which had been the primary trend at least since David Hume, who wrote in the mid-to-late 18th century). Secondly, and of more direct importance to modern method, Peirce put forth the basic schema for hypothesis/testing that continues to prevail today. Extracting the theory of inquiry from its raw materials in classical logic, he refined it in parallel with the early development of symbolic logic to address the then-current problems in scientific reasoning. Peirce examined and articulated the three fundamental modes of reasoning that, as discussed above in this article, play a role in inquiry today, the processes that are currently known as abductive, deductive, and inductive inference. Thirdly, he played a major role in the progress of symbolic logic itself â indeed this was his primary specialty.

Karl Popper denied the existence of evidence[71] and of scientific method.[72] Popper holds that there is only one universal method, the negative method of trial and error. It covers not only all products of the human mind, including science, mathematics, philosophy, art and so on, but also the evolution of life. Beginning in the 1930s he argued that empirical hypotheses must be falsifiable and that there is no such thing as inductive reasoning.[citation needed] Thus, all inferences ever made, including in science, are purely[citation needed] deductive according to Popper's view. Following Peirce and others, he argued that science is fallible and cannot be made certain or even probable[citation needed]. In contrast to empiricist-inductivist views, Popper welcomed metaphysics and philosophical discussion and even gave qualified support to myths and pseudosciences[citation needed]. Popper's view has become known as critical rationalism.

[edit] 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 can clearly 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.[73]

Mathematical work and scientific work can inspire each other.[74] 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 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,[75] the construction of mathematical proofs, and heuristic[76][77] show that the mathematical method and the scientific method differ in detail, while nevertheless resembling each other in using iterative or recursive steps.

Mathematical method Scientific method
1 Understanding Characterization from experience and observation
2 Analysis Hypothesis: a proposed explanation
3 Synthesis Deduction: prediction from the hypothesis
4 Review/Extend Test and experiment

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,[78] 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[79] 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

[edit] See also

[edit] Problems and issues

[edit] History, philosophy, sociology

[edit] Notes and references

  1. ^ Goldhaber & Nieto 2010, p. 940
  2. ^ "[4] Rules for the study of natural philosophy", Newton 1999, pp. 794â6, from Book 3, The System of the World.
  3. ^ scientific method, Merriam-Webster Dictionary.
  4. ^ Morris Kline (1985) Mathematics for the nonmathematician. Courier Dover Publications. p. 284. ISBN 0486248232
  5. ^ Peirce, C. S., Collected Papers v. 1, paragraph 74.
  6. ^ Alhazen (Ibn Al-Haytham) Critique of Ptolemy, translated by S. Pines, Actes X Congrès internationale d'histoire des sciences, Vol I Ithaca 1962, as quoted in Sambursky 1974, p. 139
  7. ^ Alhazen, translated into English from German by M. Schwarz, from "Abhandlung über das Licht", J. Baarmann (ed. 1882) Zeitschrift der Deutschen Morgenländischen Gesellschaft Vol 36 as quoted in Sambursky 1974, p. 136
  8. ^ as quoted in Sambursky 1974, p. 136
  9. ^ "...the statement of a lawâA depends on Bâalways transcends experience." âBorn 1949, p. 6
  10. ^ Taleb 2007 e.g., p. 58, devotes his chapter 5 to the error of confirmation.
  11. ^ "I believe that we do not know anything for certain, but everything probably." âChristiaan Huygens, Letter to Pierre Perrault, 'Sur la préface de M. Perrault de son traité del'Origine des fontaines' [1763], Oeuvres Complétes de Christiaan Huygens (1897), Vol. 7, 298. Quoted in Jacques Roger, The Life Sciences in Eighteenth-Century French Thought, ed. Keith R. Benson and trans. Robert Ellrich (1997), 163. Quotation selected by Bynum & Porter 2005, p. 317 Huygens 317#4.
  12. ^ As noted by Alice Calaprice (ed. 2005) The New Quotable Einstein Princeton University Press and Hebrew University of Jerusalem, ISBN 0-691-12074-9 p. 291. Calaprice denotes this not as an exact quotation, but as a paraphrase of a translation of A. Einstein's "Induction and Deduction". Collected Papers of Albert Einstein 7 Document 28. Volume 7 is The Berlin Years: Writings, 1918-1921. A. Einstein; M. Janssen, R. Schulmann, et al., eds.
  13. ^ Hempel, Carl [The Internet Encyclopedia of Philosophy]
  14. ^ Fleck 1975, pp. xxvii-xxviii
  15. ^ SCOPE - Salzberg, et al
  16. ^ October, 1951. as noted in McElheny 2004, p. 40:"That's what a helix should look like!" Crick exclaimed in delight (This is the Cochran-Crick-Vand&Stokes theory of the transform of a helix).
  17. ^ June, 1952. as noted in McElheny 2004, p. 43: Watson had succeeded in getting X-ray pictures of TMV showing a helical pattern.
  18. ^ a b Cochran W, Crick FHC and Vand V. (1952) "The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix", Acta Cryst., 5, 581-586.
  19. ^ a b Friday, January 30, 1953. Tea time. as noted in McElheny 2004, p. 52: Franklin confronts Watson and his paper - "Of course it [Pauling's pre-print] is wrong. DNA is not a helix." Watson runs away from Franklin and runs into Wilkins; they retreat to Wilkins' office, where Wilkins shows Watson photo 51. Watson immediately recognizes the diffraction pattern of a helix.
  20. ^ a b Saturday, February 28, 1953, as noted in McElheny 2004, pp. 57â59: Watson finds the base pairing which explains Chargaff's rules using his cardboard models.
  21. ^ "Observation and experiment are subject to a very popular myth. ... The knower is seen as a ... Julius Caesar winning his battles according to ... formula. Even research workers will admit that the first observation may have been a little imprecise, whereas the second and third were 'adjusted to the facts' ... until tradition, education, and familiarity have produced a readiness for stylized (that is directed and restricted) perception and action; until an answer becomes largely pre-formed in the question, and a decision confined merely to 'yes' or 'no' or perhaps to a numerical determination; until methods and apparatus automatically carry out the greatest part of the mental work for us." Ludwik Fleck labels this thought style(Denkstil). Fleck 1975, p. 84.
  22. ^ Needham & Wang 1954 p.166
  23. ^ See the hypothethico-deductive method, for example, Godfrey-Smith 2003, p. 236.
  24. ^ Jevons 1874, pp. 265â6.
  25. ^ pp.65,73,92,398 âAndrew J. Galambos, Sic Itur ad Astra ISBN 0-88078-004-5(AJG learned scientific method from Felix Ehrenhaft
  26. ^ Galilei 1638, pp. v-xii,1-300
  27. ^ Brody 1993, pp. 10â24 calls this the "epistemic cycle": "The epistemic cycle starts from an initial model; iterations of the cycle then improve the model until an adequate fit is achieved."
  28. ^ Iteration example: Chaldean astronomers such as Kidinnu compiled astronomical data. Hipparchus was to use this data to calculate the precession of the Earth's axis. Fifteen hundred years after Kidinnu, Al-Batani, born in what is now Turkey, would use the collected data and improve Hipparchus' value for the precession of the Earth's axis. Al-Batani's value, 54.5 arc-seconds per year, compares well to the current value of 49.8 arc-seconds per year (26,000 years for Earth's axis to round the circle of nutation).
  29. ^ Recursion example: the Earth is itself a magnet, with its own North and South PolesWilliam Gilbert (in Latin 1600) De Magnete, or On Magnetism and Magnetic Bodies. Translated from Latin to English, selection by Moulton & Schifferes 1960, pp. 113â117
  30. ^ "The foundation of general physics ... is experience. These ... everyday experiences we do not discover without deliberately directing our attention to them. Collecting information about these is observation." âHans Christian Ørsted("First Introduction to General Physics" â13, part of a series of public lectures at the University of Copenhagen. Copenhagen 1811, in Danish, printed by Johan Frederik Schulz. In Kirstine Meyer's 1920 edition of Ørsted's works, vol.III pp. 151-190. ) "First Introduction to Physics: the Spirit, Meaning, and Goal of Natural Science". Reprinted in German in 1822, Schweigger's Journal für Chemie und Physik 36, pp.458-488, as translated in Ørsted 1997, p. 292
  31. ^ "When it is not clear under which law of nature an effect or class of effect belongs, we try to fill this gap by means of a guess. Such guesses have been given the name conjectures or hypotheses." âHans Christian Ørsted(1811) "First Introduction to General Physics" as translated in Ørsted 1997, p. 297.
  32. ^ "In general we look for a new law by the following process. First we guess it. ...", âFeynman 1965, p. 156
  33. ^ "... the statement of a law - A depends on B - always transcends experience."âBorn 1949, p. 6
  34. ^ "The student of nature ... regards as his property the experiences which the mathematician can only borrow. This is why he deduces theorems directly from the nature of an effect while the mathematician only arrives at them circuitously." âHans Christian Ørsted(1811) "First Introduction to General Physics" â17. as translated in Ørsted 1997, p. 297.
  35. ^ Salviati speaks: "I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing ten times as much as the other, if allowed to fall, at the same instant, from a height of, say, 100 cubits, would so differ in speed that when the heavier had reached the ground, the other would not have fallen more than 10 cubits." Two New Sciences (1638) âGalilei 1638, pp. 61â62. A more extended quotation is referenced by Moulton & Schifferes 1960, pp. 80â81.
  36. ^ In the inquiry-based education paradigm, the stage of "characterization, observation, definition, â" is more briefly summed up under the rubric of a Question
  37. ^ "To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science." âEinstein & Infeld 1938, p. 92.
  38. ^ Crawford S, Stucki L (1990), "Peer review and the changing research record", "J Am Soc Info Science", vol. 41, pp 223-228
  39. ^ See, e.g., Gauch 2003, esp. chapters 5-8
  40. ^ Cartwright, Nancy (1983), How the Laws of Physics Lie. Oxford: Oxford University Press. ISBN 0-19-824704-4
  41. ^ Andreas Vesalius, Epistola, Rationem, Modumque Propinandi Radicis Chynae Decocti (1546), 141. Quoted and translated in C.D. O'Malley, Andreas Vesalius of Brussels, (1964), 116. As quoted by Bynum & Porter 2005, p. 597: Andreas Vesalius,597#1.
  42. ^ Crick, Francis (1994), The Astonishing Hypothesis ISBN 0-684-19431-7 p.20
  43. ^ Glen 1994, pp. 37â38.
  44. ^ "The structure that we propose is a three-chain structure, each chain being a helix" â Linus Pauling, as quoted on p. 157 by Horace Freeland Judson (1979), The Eighth Day of Creation ISBN 0-671-22540-5
  45. ^ McElheny 2004, pp. 49â50: January 28, 1953 - Watson read Pauling's pre-print, and realized that in Pauling's model, DNA's phosphate groups had to be un-ionized. But DNA is an acid, which contradicts Pauling's model.
  46. ^ June, 1952. as noted in McElheny 2004, p. 43: Watson had succeeded in getting X-ray pictures of TMV showing a diffraction pattern consistent with the transform of a helix.
  47. ^ Watson did enough work on Tobacco mosaic virus to produce the diffraction pattern for a helix, per Crick's work on the transform of a helix. pp. 137-138, Horace Freeland Judson (1979) The Eighth Day of Creation ISBN 0-671-22540-5
  48. ^ In March 1917, the Royal Astronomical Society announced that on May 29, 1919, the occasion of a total eclipse of the sun would afford favorable conditions for testing Einstein's General theory of relativity. One expedition, to Sobral, Cear¡, Brazil, and Eddington's expedition to the island of Principe yielded a set of photographs, which, when compared to photographs taken at Sobral and at Greenwich Observatory showed that the deviation of light was measured to be 1.69 arc-seconds, as compared to Einstein's desk prediction of 1.75 arc-seconds. â Antonina Vallentin (1954), Einstein, as quoted by Samuel Rapport and Helen Wright (1965), Physics, New York: Washington Square Press, pp 294-295.
  49. ^ "The instant I saw the picture my mouth fell open and my pulse began to race." âWatson 1968, p. 167 Page 168 shows the X-shaped pattern of the B-form of DNA, clearly indicating crucial details of its helical structure to Watson and Crick.
  50. ^ "Suddenly I became aware that an adenine-thymine pair held together by two hydrogen bonds was identical in shape to a guanine-cytosine pair held together by at least two hydrogen bonds. ..." âWatson 1968, pp. 194â197.
  51. ^ See, e.g., Physics Today, 59(1), p42. Richmann electrocuted in St. Petersburg (1753)
  52. ^ Aristotle, "Prior Analytics", Hugh Tredennick (trans.), pp. 181-531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
  53. ^ a b Peirce, C.S. (1877), "The Fixation of Belief", Popular Science Monthly, v. 12, pp. 1â15. Reprinted often, including (Collected Papers of Charles Sanders Peirce v. 5, paragraphs 358â87), (The Essential Peirce, v. 1, pp. 109â23). Peirce.org Eprint. Wikisource Eprint.
  54. ^ Peirce, C.S. (1898), "Philosophy and the Conduct of Life", Lecture 1 of the Cambridge (MA) Conferences Lectures, published in Collected Papers v. 1, paragraphs 616-48 in part and in Reasoning and the Logic of Things, Ketner (ed., intro.) and Putnam (intro., comm.), pp. 105-22, reprinted in The Essential Peirce v. 2, pp. 27-41.
  55. ^ Peirce (1899), "F.R.L." [First Rule of Logic], Collected Papers v. 1, paragraphs 135-40, Eprint
  56. ^ Collected Papers v. 5, in paragraph 582, from 1898:
    ... [scientific] inquiry of every type, fully carried out, has the vital power of self-correction and of growth. This is a property so deeply saturating its inmost nature that it may truly be said that there is but one thing needful for learning the truth, and that is a hearty and active desire to learn what is true.
  57. ^ a b c Peirce, C.S. (1877), "How to Make Our Ideas Clear", Popular Science Monthly, v. 12, pp. 286â302. Reprinted often, including (Collected Papers v. 5, paragraphs 388â410), (The Essential Peirce v. 1, pp. 124â41). Arisbe Eprint, Wikisource Eprint.
  58. ^ Peirce, C.S. (1868), "Some Consequences of Four Incapacities", Journal of Speculative Philosophy v. 2, n. 3, pp. 140â57. Reprinted often, including (Collected Papers v. 5, paragraphs 264â317), (The Essential Peirce v. 1, pp. 28â55). Arisbe Eprint
  59. ^ Peirce, C.S. (1878), "The Doctrine of Chances", Popular Science Monthly v. 12, pp. 604-15, see pp. 610-11 via Internet Archive. Reprinted (Collected Papers v. 2, paragraphs 645-68), (The Essential Peirce v. 1, pp. 142-54). "...death makes the number of our risks, the number of our inferences, finite, and so makes their mean result uncertain. The very idea of probability and of reasoning rests on the assumption that this number is indefinitely great. .... ...logicality inexorably requires that our interests shall not be limited. .... Logic is rooted in the social principle."
  60. ^ a b c Peirce, C.S. (1908), "A Neglected Argument for the Reality of God", Hibbert Journal v. 7, pp. 90-112. Wikisource Eprint with added notes. Reprinted often, with previously unpublished part, including (Collected Papers v. 6, paragraphs 452-85), (The Essential Peirce v. 2, pp. 434-50).
  61. ^ Peirce, C.S. (1902), application to the Carnegie Institution, see MS L75.329-330, from Draft D of Memoir 27:
    Consequently, to discover is simply to expedite an event that would occur sooner or later, if we had not troubled ourselves to make the discovery. Consequently, the art of discovery is purely a question of economics. The economics of research is, so far as logic is concerned, the leading doctrine with reference to the art of discovery. Consequently, the conduct of abduction, which is chiefly a question of heuretic and is the first question of heuretic, is to be governed by economical considerations.
  62. ^ Peirce (c. 1906), "PAP (Prolegomena for an Apology to Pragmatism)" (Manuscript 293, not the like-named article), The New Elements of Mathematics (NEM) 4:319-320, see first quote under "Abduction" at Commens Dictionary of Peirce's Terms.
  63. ^ Peirce, C.S. (1903), "Pragmatism â The Logic of Abduction", Collected Papers v. 5, paragraphs 195-205, especially 196. Eprint.
  64. ^ Peirce, C.S., (1878) "The Probability of Induction", Popular Science Monthly, v. 12, pp. 705-18, Google Books Eprint, Internet Archive Eprint. Reprinted often, including (Collected Papers v. 2, paragraphs 669-93), (The Essential Peirce v. 1, pp. 155-69).
  65. ^
    • Higher Superstition: The Academic Left and Its Quarrels with Science, The Johns Hopkins University Press, 1997
    • Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science, Picador; 1st Picador USA Pbk. Ed edition, 1999
    • The Sokal Hoax: The Sham That Shook the Academy, University of Nebraska Press, 2000 ISBN 0-8032-7995-7
    • A House Built on Sand: Exposing Postmodernist Myths About Science, Oxford University Press, 2000
    • Intellectual Impostures, Economist Books, 2003
  66. ^ The ancient Egyptians observed that heliacal rising of a certain star, Sothis (Greek for Sopdet (Egyptian), known to the West as Sirius), marked the annual flooding of the Nile river. See Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (2 ed.), Dover Publications, ISBN 978-048622332-2, http://books.google.com/?id=JVhTtVA2zr8C , p.82, and also the 1911 Britannica, "Egypt".
  67. ^ The Rhind papyrus lists practical examples in arithmetic and geometry â1911 Britannica, "Egypt".
  68. ^ The Ebers papyrus lists some of the 'mysteries of the physician', as cited in the 1911 Britannica, "Egypt"
  69. ^ Rosanna Gorini (2003), "Al-Haytham the Man of Experience, First Steps in the Science of Vision", International Society for the History of Islamic Medicine, Institute of Neurosciences, Laboratory of Psychobiology and Psychopharmacology, Rome, Italy:
    "According to the majority of the historians al-Haytham was the pioneer of the modern scientific method. With his book he changed the meaning of the term optics and established experiments as the norm of proof in the field. His investigations are based not on abstract theories, but on experimental evidences and his experiments were systematic and repeatable."
  70. ^ "John Stuart Mill (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. http://plato.stanford.edu/entries/mill/#SciMet. Retrieved 2009-07-31. 
  71. ^ Logik der Forschung, new appendix *XIX (not yet available in the English edition Logic of scientific discovery)
  72. ^ Karl Popper: On the non-existence of scientific method. Realism and the Aim of Science (1983)
  73. ^ "When we are working intensively, we feel keenly the progress of our work; we are elated when our progress is rapid, we are depressed when it is slow." â the mathematician Pólya 1957, p. 131 in the section on 'Modern heuristic'.
  74. ^ "Philosophy [i.e., physics] is written in this grand book--I mean the universe--which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth." âGalileo Galilei, Il Saggiatore (The Assayer, 1623), as translated by Stillman Drake (1957), Discoveries and Opinions of Galileo pp. 237-8, as quoted by di Francia 1981, p. 10.
  75. ^ Pólya 1957 2nd ed.
  76. ^ George Pólya (1954), Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics,
  77. ^ George Pólya (1954), Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Reasoning.
  78. ^ Pólya 1957, p. 142
  79. ^ Pólya 1957, p. 144
  80. ^ Mackay 1991 p.100

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