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Hypothesis, Model, Theory, and Law

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In common usage, the words hypothesis, model, theory, and law have different interpretations and are at times used without precision, but in science they have very exact meanings.

Perhaps the most difficult and intriguing step is the development of a specific, testable hypothesis. A useful hypothesis enables predictions by applying deductive reasoning, often in the form of mathematical analysis. It is a limited statement regarding the cause and effect in a specific situation, which can be tested by experimentation and observation or by statistical analysis of the probabilities from the data obtained. The outcome of the test hypothesis should be currently unknown, so that the results can provide useful data regarding the validity of the hypothesis.

Sometimes a hypothesis is developed that must wait for new knowledge or technology to be testable. The concept of atoms was proposed by the ancient Greeks , who had no means of testing it. Centuries later, when more knowledge became available, the hypothesis gained support and was eventually accepted by the scientific community, though it has had to be amended many times over the year. Atoms are not indivisible, as the Greeks supposed.

A model is used for situations when it is known that the hypothesis has a limitation on its validity. The Bohr model of the atom , for example, depicts electrons circling the atomic nucleus in a fashion similar to planets in the solar system. This model is useful in determining the energies of the quantum states of the electron in the simple hydrogen atom, but it is by no means represents the true nature of the atom. Scientists (and science students) often use such idealized models  to get an initial grasp on analyzing complex situations.

Theory and Law

A scientific theory or law represents a hypothesis (or group of related hypotheses) which has been confirmed through repeated testing, almost always conducted over a span of many years. Generally, a theory is an explanation for a set of related phenomena, like the theory of evolution or the big bang theory . 

The word "law" is often invoked in reference to a specific mathematical equation that relates the different elements within a theory. Pascal's Law refers an equation that describes differences in pressure based on height. In the overall theory of universal gravitation developed by Sir Isaac Newton , the key equation that describes the gravitational attraction between two objects is called the law of gravity .

These days, physicists rarely apply the word "law" to their ideas. In part, this is because so many of the previous "laws of nature" were found to be not so much laws as guidelines, that work well within certain parameters but not within others.

Scientific Paradigms

Once a scientific theory is established, it is very hard to get the scientific community to discard it. In physics, the concept of ether as a medium for light wave transmission ran into serious opposition in the late 1800s, but it was not disregarded until the early 1900s, when Albert Einstein proposed alternate explanations for the wave nature of light that did not rely upon a medium for transmission.

The science philosopher Thomas Kuhn developed the term scientific paradigm to explain the working set of theories under which science operates. He did extensive work on the scientific revolutions that take place when one paradigm is overturned in favor of a new set of theories. His work suggests that the very nature of science changes when these paradigms are significantly different. The nature of physics prior to relativity and quantum mechanics is fundamentally different from that after their discovery, just as biology prior to Darwin’s Theory of Evolution is fundamentally different from the biology that followed it. The very nature of the inquiry changes.

One consequence of the scientific method is to try to maintain consistency in the inquiry when these revolutions occur and to avoid attempts to overthrow existing paradigms on ideological grounds.

Occam’s Razor

One principle of note in regards to the scientific method is Occam’s Razor (alternately spelled Ockham's Razor), which is named after the 14th century English logician and Franciscan friar William of Ockham. Occam did not create the concept—the work of Thomas Aquinas and even Aristotle referred to some form of it. The name was first attributed to him (to our knowledge) in the 1800s, indicating that he must have espoused the philosophy enough that his name became associated with it.

The Razor is often stated in Latin as:

entia non sunt multiplicanda praeter necessitatem

or, translated to English:

entities should not be multiplied beyond necessity

Occam's Razor indicates that the most simple explanation that fits the available data is the one which is preferable. Assuming that two hypotheses presented have equal predictive power, the one which makes the fewest assumptions and hypothetical entities takes precedence. This appeal to simplicity has been adopted by most of science, and is invoked in this popular quote by Albert Einstein:

Everything should be made as simple as possible, but not simpler.

It is significant to note that Occam's Razor does not prove that the simpler hypothesis is, indeed, the true explanation of how nature behaves. Scientific principles should be as simple as possible, but that's no proof that nature itself is simple.

However, it is generally the case that when a more complex system is at work there is some element of the evidence which doesn't fit the simpler hypothesis, so Occam's Razor is rarely wrong as it deals only with hypotheses of purely equal predictive power. The predictive power is more important than the simplicity.

Edited by Anne Marie Helmenstine, Ph.D.

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This is the Difference Between a Hypothesis and a Theory

What to Know A hypothesis is an assumption made before any research has been done. It is formed so that it can be tested to see if it might be true. A theory is a principle formed to explain the things already shown in data. Because of the rigors of experiment and control, it is much more likely that a theory will be true than a hypothesis.

As anyone who has worked in a laboratory or out in the field can tell you, science is about process: that of observing, making inferences about those observations, and then performing tests to see if the truth value of those inferences holds up. The scientific method is designed to be a rigorous procedure for acquiring knowledge about the world around us.

In scientific reasoning, a hypothesis is constructed before any applicable research has been done. A theory, on the other hand, is supported by evidence: it's a principle formed as an attempt to explain things that have already been substantiated by data.

Toward that end, science employs a particular vocabulary for describing how ideas are proposed, tested, and supported or disproven. And that's where we see the difference between a hypothesis and a theory .

A hypothesis is an assumption, something proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

What is a Hypothesis?

A hypothesis is usually tentative, an assumption or suggestion made strictly for the objective of being tested.

When a character which has been lost in a breed, reappears after a great number of generations, the most probable hypothesis is, not that the offspring suddenly takes after an ancestor some hundred generations distant, but that in each successive generation there has been a tendency to reproduce the character in question, which at last, under unknown favourable conditions, gains an ascendancy. Charles Darwin, On the Origin of Species , 1859 According to one widely reported hypothesis , cell-phone transmissions were disrupting the bees' navigational abilities. (Few experts took the cell-phone conjecture seriously; as one scientist said to me, "If that were the case, Dave Hackenberg's hives would have been dead a long time ago.") Elizabeth Kolbert, The New Yorker , 6 Aug. 2007

What is a Theory?

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, its likelihood as truth is much higher than that of a hypothesis.

It is evident, on our theory , that coasts merely fringed by reefs cannot have subsided to any perceptible amount; and therefore they must, since the growth of their corals, either have remained stationary or have been upheaved. Now, it is remarkable how generally it can be shown, by the presence of upraised organic remains, that the fringed islands have been elevated: and so far, this is indirect evidence in favour of our theory . Charles Darwin, The Voyage of the Beagle , 1839 An example of a fundamental principle in physics, first proposed by Galileo in 1632 and extended by Einstein in 1905, is the following: All observers traveling at constant velocity relative to one another, should witness identical laws of nature. From this principle, Einstein derived his theory of special relativity. Alan Lightman, Harper's , December 2011

Non-Scientific Use

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch (though theory is more common in this regard):

The theory of the teacher with all these immigrant kids was that if you spoke English loudly enough they would eventually understand. E. L. Doctorow, Loon Lake , 1979 Chicago is famous for asking questions for which there can be no boilerplate answers. Example: given the probability that the federal tax code, nondairy creamer, Dennis Rodman and the art of mime all came from outer space, name something else that has extraterrestrial origins and defend your hypothesis . John McCormick, Newsweek , 5 Apr. 1999 In his mind's eye, Miller saw his case suddenly taking form: Richard Bailey had Helen Brach killed because she was threatening to sue him over the horses she had purchased. It was, he realized, only a theory , but it was one he felt certain he could, in time, prove. Full of urgency, a man with a mission now that he had a hypothesis to guide him, he issued new orders to his troops: Find out everything you can about Richard Bailey and his crowd. Howard Blum, Vanity Fair , January 1995

And sometimes one term is used as a genus, or a means for defining the other:

Laplace's popular version of his astronomy, the Système du monde , was famous for introducing what came to be known as the nebular hypothesis , the theory that the solar system was formed by the condensation, through gradual cooling, of the gaseous atmosphere (the nebulae) surrounding the sun. Louis Menand, The Metaphysical Club , 2001 Researchers use this information to support the gateway drug theory — the hypothesis that using one intoxicating substance leads to future use of another. Jordy Byrd, The Pacific Northwest Inlander , 6 May 2015 Fox, the business and economics columnist for Time magazine, tells the story of the professors who enabled those abuses under the banner of the financial theory known as the efficient market hypothesis . Paul Krugman, The New York Times Book Review , 9 Aug. 2009

Incorrect Interpretations of "Theory"

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general use to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

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What are the differences between hypothesis, supposition, assumption, postulate, and axiom?

• 1 No difference in colloquial language. In logic, it seems to me, supposition is not usually used. –  Mauro ALLEGRANZA Commented Apr 7, 2017 at 21:18
• This seems like a question about definition/use of terms, which is not a question for this SE. The question also lacks context, is there some philosophical background to it? –  Conifold Commented Apr 7, 2017 at 21:23
• @Conifold in other words, maybe this question is more appropriate for Mathematics or History of Science? –  DukeZhou Commented Apr 7, 2017 at 23:28
• 1 @DukeZhou Well, Math SE already has Difference between axioms, theorems, postulates, corollaries, and hypotheses . Acerbi goes deep into the original distinctions between axioms and postulates in Aristotle and Euclid. But I am not sure if Geremia is interested in mathematical use or something else. –  Conifold Commented Apr 7, 2017 at 23:44
• 1 As a rule of thumb, the usage is: assumption and hypothesis are "starting points" of some argument, "assumed" as true or reasonable for the context of the argument, while postulate and axiom are more related to a theory, and thus "assumed" as true as long as we maintain the corresponding theory. –  Mauro ALLEGRANZA Commented Apr 8, 2017 at 9:16

Heath, the famous translator of the Elements , concludes in his introduction to vol. 1 of his translation of the Elements , §"3. First Principles: Definitions , Postulates , and Axioms ", that Euclid's usage of these terms aligns most closely to Aristotle's. Heath begins that § by quoting in extenso from Aristotle's Posterior Analytics 1.10 (76 a 5) ("Difference between principles and non-principles, common and proper principles") and commenting upon it. Here's Heath's translation with his useful parenthetical remarks relating what Aristotle is saying to geometry:

“By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence , this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers. “Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate . For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis , not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate . This is the difference between a hypothesis and a postulate ; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses , for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis , unless indeed it be asserted that any audible speech is a hypothesis . A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer’s hypotheses false, as some have said: I mean those who say that ’you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.’ The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither” (because the subject is of equal extent with what is predicated of it).

source: my answer to HSM.SE's " How did Aristotle influence Euclid? ", with extra emboldening of terms

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scientific hypothesis

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• National Center for Biotechnology Information - PubMed Central - On the scope of scientific hypotheses
• LiveScience - What is a scientific hypothesis?
• The Royal Society - Open Science - On the scope of scientific hypotheses

scientific hypothesis , an idea that proposes a tentative explanation about a phenomenon or a narrow set of phenomena observed in the natural world. The two primary features of a scientific hypothesis are falsifiability and testability, which are reflected in an “If…then” statement summarizing the idea and in the ability to be supported or refuted through observation and experimentation. The notion of the scientific hypothesis as both falsifiable and testable was advanced in the mid-20th century by Austrian-born British philosopher Karl Popper .

The formulation and testing of a hypothesis is part of the scientific method , the approach scientists use when attempting to understand and test ideas about natural phenomena. The generation of a hypothesis frequently is described as a creative process and is based on existing scientific knowledge, intuition , or experience. Therefore, although scientific hypotheses commonly are described as educated guesses, they actually are more informed than a guess. In addition, scientists generally strive to develop simple hypotheses, since these are easier to test relative to hypotheses that involve many different variables and potential outcomes. Such complex hypotheses may be developed as scientific models ( see scientific modeling ).

Depending on the results of scientific evaluation, a hypothesis typically is either rejected as false or accepted as true. However, because a hypothesis inherently is falsifiable, even hypotheses supported by scientific evidence and accepted as true are susceptible to rejection later, when new evidence has become available. In some instances, rather than rejecting a hypothesis because it has been falsified by new evidence, scientists simply adapt the existing idea to accommodate the new information. In this sense a hypothesis is never incorrect but only incomplete.

The investigation of scientific hypotheses is an important component in the development of scientific theory . Hence, hypotheses differ fundamentally from theories; whereas the former is a specific tentative explanation and serves as the main tool by which scientists gather data, the latter is a broad general explanation that incorporates data from many different scientific investigations undertaken to explore hypotheses.

Countless hypotheses have been developed and tested throughout the history of science . Several examples include the idea that living organisms develop from nonliving matter, which formed the basis of spontaneous generation , a hypothesis that ultimately was disproved (first in 1668, with the experiments of Italian physician Francesco Redi , and later in 1859, with the experiments of French chemist and microbiologist Louis Pasteur ); the concept proposed in the late 19th century that microorganisms cause certain diseases (now known as germ theory ); and the notion that oceanic crust forms along submarine mountain zones and spreads laterally away from them ( seafloor spreading hypothesis ).

The Scientific Meaning of the Terms

Lay people often misinterpret the language used by scientists. And for that reason, they sometimes draw the wrong conclusions as to what the scientific terms mean.

Three such terms that are often used interchangeably are “scientific law,” “hypothesis,” and  “theory.”

In layman’s terms, if something is said to be “just a theory,” it usually means that it is a mere guess, or is unproved. It might even lack credibility. But in scientific terms, a theory implies that something has been proven and is generally accepted as being true.

Here is what each of these terms means to a scientist:

This is a statement of fact meant to describe, in concise terms, an action or set of actions. It is generally accepted to be true and universal, and can sometimes be expressed in terms of a single mathematical equation. Scientific laws are similar to mathematical postulates. They don’t really need any complex external proofs; they are accepted at face value based upon the fact that they have always been observed to be true.

Specifically, scientific laws must be simple, true, universal, and absolute. They represent the cornerstone of scientific discovery, because if a law ever did not apply, then all science based upon that law would collapse.

Some scientific laws, or laws of nature, include the law of gravity, Newton’s laws of motion, the laws of thermodynamics, Boyle’s law of gases, the law of conservation of mass and energy, and Hook’s law of elasticity.

Hypothesis:

This is an educated guess based upon observation. It is a rational explanation of a single event or phenomenon based upon what is observed, but which has not been proved. Most hypotheses can be supported or refuted by experimentation or continued observation.

Scientific Theory:

A theory is what one or more hypotheses become once they have been verified and accepted to be true. A theory is an explanation of a set of related observations or events based upon proven hypotheses and verified multiple times by detached groups of researchers. Unfortunately, even some scientists often use the term “theory” in a more colloquial sense, when they really mean to say “hypothesis.” That makes its true meaning in science even more confusing to the general public.

In general, both a scientific theory and a scientific law are accepted to be true by the scientific community as a whole. Both are used to make predictions of events. Both are used to advance technology.

In fact, some laws, such as the law of gravity, can also be theories when taken more generally. The law of gravity is expressed as a single mathematical expression and is presumed to be true all over the universe and all through time. Without such an assumption, we can do no science based on gravity’s effects. But from the law, we derived the theory of gravity which describes how gravity works, what causes it, and how it behaves. We also use that to develop another theory, Einstein’s General Theory of Relativity, in which gravity plays a crucial role. The basic law is intact, but the theory expands it to include various and complex situations involving space and time.

The biggest difference between a law and a theory is that a theory is much more complex and dynamic. A law describes a single action, whereas a theory explains an entire group of related phenomena. And, whereas a law is a postulate that forms the foundation of the scientific method, a theory is the end result of that same process.

A simple analogy can be made using a slingshot and an automobile.

A scientific law is like a slingshot. A slingshot has but one moving part–the rubber band. If you put a rock in it and draw it back, the rock will fly out at a predictable speed, depending upon the distance the band is drawn back.

An automobile has many moving parts, all working in unison to perform the chore of transporting someone from one point to another point. An automobile is a complex piece of machinery. Sometimes, improvements are made to one or more component parts. A new set of spark plugs that are composed of a better alloy that can withstand heat better, for example, might replace the existing set. But the function of the automobile as a whole remains unchanged.

A theory is like the automobile. Components of it can be changed or improved upon, without changing the overall truth of the theory as a whole.

Some scientific theories include the theory of evolution, the theory of relativity, the atomic theory, and the quantum theory. All of these theories are well documented and proved beyond reasonable doubt. Yet scientists continue to tinker with the component hypotheses of each theory in an attempt to make them more elegant and concise, or to make them more all-encompassing. Theories can be tweaked, but they are seldom, if ever, entirely replaced.

A theory is developed only through the scientific method, meaning it is the final result of a series of rigorous processes. Note that theories do not become laws. Scientific laws must exist prior to the start of using the scientific method because, as stated earlier, laws are the foundation for all science. Here is an oversimplified example of the development of a scientific theory:

Development of a Simple Theory by the Scientific Method:

• Start with an observation that evokes a question:  Broth spoils when I leave it out for a couple of days. Why?
• Using logic and previous knowledge, state a possible answer, called a Hypothesis:  Tiny organisms floating in the air must fall into the broth and start reproducing.
• Perform an experiment or Test:  After boiling some broth, I divide it into two containers, one covered and one not covered. I place them on the table for two days and see if one spoils. Only the uncovered broth spoiled.
• Then publish your findings in a peer-reviewed journal. Publication:  “Only broth that is exposed to the air after two days tended to spoil. The covered specimen did not.”
• Other scientists read about your experiment and try to duplicate it. Verification:  Every scientist who tries your experiment comes up with the same results. So they try other methods to make sure your experiment was measuring what it was supposed to. Again, they get the same results every time.
• In time, and if experiments continue to support your hypothesis, it becomes a Scientific Theory:  Microorganisms from the air cause broth to spoil.

Useful Prediction:

If I leave food items open to the air, they will spoil. If I want to keep them from spoiling, I will keep them covered.

Note, however, that although the prediction is useful, the theory does not absolutely  prove  that the next open container of broth will spoil. Thus it is said to be falsifiable. If anyone ever left a cup of broth open for days and it did not spoil, the theory would have to be tweaked or thrown out.

A real theory must be falsifiable. They must be capable of being modified based on new evidence. So-called “theories” based on religion, such as creationism or intelligent design are, therefore, not scientific theories. They are not falsifiable, they don’t depend on new evidence, and they do not follow the scientific method.

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"Hypothesize" vs "postulate"

When writing a scientific or engineering paper, how do we choose between hypothesize and postulate ?

• writing-style

• That question seems like it would be best asked of a more local scientifically oriented site, with its own cultural preferences. You should probably rewrite your question so that it is purely about hypothesize/postulate. –  Mitch Commented Jan 5, 2012 at 14:08
• 1 @Mitch But postulation outside of a scientific paper sounds a bit pretentious. –  z7sg Ѫ Commented Jan 5, 2012 at 14:38
• 1 @z7sgѪ If you mean postulate, you should say postulate. If someone wants to think you're pretentious, so be it. There isn't really an exact synonym. I mean, you could use assume , but it would be imprecise. –  slim Commented Jan 5, 2012 at 15:10

4 Answers 4

When you postulate , you're saying "let's all agree for the purposes of this discussion that (something) is true."

When you hypothesise , you're saying "Let's speculate about what would happen if (something) was true".

A hypothesis has some extra status in scientific discourse, in that scientists frequently put forward hypotheses they consider to be plausible, and perform tests to see whether they stand up to them.

• 1 to hypothesize something (and this is generally how the word is used in academic papers) means roughly the same as to postulate something . At least as far as I can see it does. Perhaps though there is some difference between the two? –  z7sg Ѫ Commented Jan 5, 2012 at 16:02
• 1 @z7sgѪ No, as my answer says, hypothesising says "what if this were true?"; postulating says "let's take this as true". –  slim Commented Jan 5, 2012 at 16:07
• I mean that in papers it's mostly stuff like: "we hypothesize that beaver dams increase the complexity of storm response..." I did think initially that you test a hypothesis and assume a postulate but looking at actual usage I'm not sure (otherwise I would have answered the question). –  z7sg Ѫ Commented Jan 5, 2012 at 16:14
• @z7sgѪ Yep, that's the use described in my third paragraph. If they said "We postulate that beaver dams...", that would mean "we think that's an uncontroversial claim that every reader can agree on without further discussion". –  slim Commented Jan 5, 2012 at 16:16

Hypothesis is a theory which can after testing be accepted or rejected. A postulate is something that is assumed to be true without proof. Sometimes postulates are also called axioms.

A hypothesis is a posed statement one wants to prove.

If it has been proved it becomes a theorem. If it has been disproved it will be discarded.

A postulate is a posed statement one doesn't want to prove.

It is used to derive other statements.

A postulate is like the opening bid for cognition--you suggest to yourself, or your scientific research group, a beginning concept for a range of phenomena with a "let's wait and see what follows from the concept" attitude of openness. Further conceptual development then follows until convergence occurs and testing for alignment with experience can begin.

An hypothesis is the outcome of the postulate, signifying a cognitional convergence and readiness for testing of the convergence with as wide a range of experience as possible.

The Greek word hypo means under. Usage as a prefix in the word hypo-thesis suggests that an hypothesis does not rise to the credibility afforded a thesis of work which has been exhaustively examined for credibility and correctness. An hypothesis may thus be regarded as a collection of inferences and conclusions , deduced as the result of an initial proposal or postulate .

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Is there a difference between a postulate and a principle in physics?

Both seem unproved statements taken as true. If thats correct, why the different names?

• terminology

• $\begingroup$ Related: physics.stackexchange.com/q/44706/2451 and links therein. $\endgroup$ –  Qmechanic ♦ Commented Aug 27, 2013 at 7:03

A postulate is an (usually fundamental) assumption a writer makes in order to discuss a subject in a coherent fashion. Examples of postulates are the Born rule in quantum mechanics (which defines how the wave function is to be interpreted), or in classical mechanics the existence of a Lagrangian (which defines the starting point of theoretical mechanics).

A principle is a more or less universally observed (usually fundamental) fact. Examples of principles are the second law of thermodynamics (universal dissipation), the principle of relativity (independence of the reference frame), or Heisenberg's uncertainty relation.

A hypothesis is a theoretical assumption made to develop a (usually alternative) theory. Examples are Planck's and Einstein's hypothesis of quantized light, or the existence of supersymmetry.

One can turn a principle or hypothesis into a postulate, but not a postulate into a principle.

Edit2: Note that it is possible that a principle is derived from a set of postulates. This reflects the fact that there is is some freedom in setting up the foundations. For example, the second law of thermodynamics can be derived from statistical mechnaics, and the principle of relativity can be derived from the postulate of Lorentz invariance.

• $\begingroup$ Huh, is not the experimental verification of first only theoretical assumptions some kind of an example of turning a postulate into an (observed) principle? I mean things like the observation of first only theoretically predicted antimatter for example. $\endgroup$ –  Dilaton Commented Sep 5, 2012 at 9:11
• $\begingroup$ @Dilaton: Not every general observable fact is referred to as a principle. I never saw the existence of antimatter formulated as principle. It is always deduced from theory. $\endgroup$ –  Arnold Neumaier Commented Sep 5, 2012 at 9:53
• $\begingroup$ Ah ok, thanks for the clarification. Seems I confused principles and theoretical predictions ... What is generally the difference between a principle and theoretical assumptions, such as say for example an additional symmetry that nature could have ? $\endgroup$ –  Dilaton Commented Sep 5, 2012 at 10:36
• 1 $\begingroup$ @Dilaton: I expanded my answer to address this. $\endgroup$ –  Arnold Neumaier Commented Sep 5, 2012 at 11:42
• $\begingroup$ Thanks Arnold. Is it ok to call a deduced idea by 'principle'? Like Heisenberg uncertainty relation. $\endgroup$ –  galmeida Commented Sep 6, 2012 at 1:52

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Terminology: Difference between Lemma, Theorem, Definition, Hypothesis, Postulate and a Proposition [duplicate]

Based on observation after reading few books and papers, I think that

Lemma : Lemma contains some information that is commonly used to support a theorem. So, a Lemma introduces a Theorem and comes before that Theorem. The information contained in the Lemma is generally used in the proof of the Theorem.

Q1: What is a Proposition ?

Q2: What is the difference between Proposition and Theorem ? A Proposition can also be proved, in the same way as a Theorem is proven.

Hypothesis : A hypothesis is like a statement for a guess, and we need to prove that analytically or experimentally.

Q3: What is the difference between Theorem and Hypothesis , for example Null hypothesis in statistics? In general, if a Theorem is always proven to be true then it no longer becomes a Hypothesis? Am I correct?

Q4: What is the difference between Postulate and Theorem ?

Q5 : What is the difference between a proposition and a definition ?

Looking for easy to remember answers. Thank you for help.

• terminology

• 1 $\begingroup$ A definition is just a 'naming' of an object and not like a theorem at all... a proposition is a theorem... usually an 'easy theorem' setting out some basic facts. $\endgroup$ –  JP McCarthy Commented Apr 22, 2015 at 17:38

I'm not the authority on this, but this is how I interpret all of these words in math literature:

Definition - This is an assignment of language and syntax to some property of a set, function, or other object. A definition is not something you prove, it is something someone assigns. Often you will want to prove that something satisfies a definition. Example: We call a mapping $f:X\to Y$ injective if whenever $f(x) = f(y)$ then $x=y$.

Proposition - This is a property that one can derive easily or directly from a given definition of an object. Example: the identity element in a group is unique.

Lemma - This is a property that one can derive or prove which is usually technical in nature and is not of primary importance to the overall body of knowledge one is trying to develop. Usually lemmas are there as precursors to larger results that one wants to obtain, or introduce a new technique or tool that one can use over and over again. Example: In a Hausdorff space, compact subsets can be separated by disjoint open subsets.

Theorem - This is a property of major importance that one can derive which usually has far-sweeping consequences for the area of math one is studying. Theorems don't necessarily need the support of propositions or lemmas, but they often do require other smaller results to support their evidence. Example: Every manifold has a simply connected covering space.

Corollary - This is usually a result that is a direct consequence of a major theorem. Often times a theorem lends itself to other smaller results or special cases which can be shown by simpler methods once a theorem is proven. Example: A consequence to the Hopf-Rinow theorem is that compact manifolds are geodesically complete.

Conjecture - This is an educated prediction that one makes based on their experience. The difference between a conjecture and a lemma/theorem/corollary is that it is usually an open research problem that either has no answer, or some partial answer. Conjectures are usually only considered important if they are authored by someone well-known in their respective area of mathematics. Once it is proven or disproven, it ceases to be a conjecture and either becomes a fact (backed by a theorem) or there is some interesting counterexample to demonstrate how it is wrong. Example: The Poincar$\acute{\text{e}}$ conjecture was a famous statement that remained an open research problem in topology for roughly a century. The claim was that every simply connected, compact 3-manifold was homeomorphic to the 3-sphere $\mathbb{S}^3$. This statement however is no longer a conjecture since it was famously proven by Grigori Perelman in 2003.

Postulate - I would appreciate community input on this, but I haven't seen this word used in any of the texts/papers I read. I would assume that this is synonymous with proposition.

• 5 $\begingroup$ I know Postulate is a synonym of axiom. Very used word in italian, but more in physics than mathematics. See wikipedia en.wikipedia.org/wiki/Axiom $\endgroup$ –  user3621272 Commented Apr 23, 2015 at 9:24
• $\begingroup$ @Mnifldz and user3621272: Thank you very much for the answers; it is easy to understand and clear. $\endgroup$ –  SKM Commented Apr 23, 2015 at 17:01

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Postulate vs. Hypothesis — What's the Difference?

Difference Between Postulate and Hypothesis

Table of contents, key differences, comparison chart, verification, compare with definitions, common curiosities, is a postulate always true, what distinguishes a postulate from a hypothesis, what role does experimentation play in testing hypotheses, how does a hypothesis contribute to scientific research, are all mathematical statements postulates, what makes a good hypothesis, why are postulates important in mathematics, can a hypothesis become a postulate, can a hypothesis be proven, what happens if a postulate is found inconsistent, why can't a hypothesis be a foundational assumption like a postulate, how do postulates and hypotheses differ in their application, what is the significance of hypotheses in developing theories, can the failure to disprove a hypothesis be considered a success, how do scientists respond when a hypothesis is disproved, share your discovery.

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