Fall Asleep While Learning About Hypotheses

In this episode of the I Can't Sleep Podcast, fall asleep while learning about hypotheses. Wait, is that too science-y for you? Well, don't worry; I included some math and other subjects you won't remember. Happy sleeping!

Transcript

Welcome to the I Can't Sleep Podcast,

Where I read random articles from across the web to bore you to sleep with my soothing voice.

I'm your host,

Benjamin Boster.

Today's episode is from a Wikipedia article titled,

Hypothesis.

A hypothesis,

Plural hypotheses,

Is a proposed explanation for a phenomenon.

For a hypothesis to be a scientific hypothesis,

A scientific method requires that one can test it.

Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories.

Even though the words hypothesis and theory are often used interchangeably,

A scientific hypothesis is not the same as a scientific theory.

A working hypothesis is a provisionally accepted hypothesis proposed for further research in a process beginning with an educated guess or thought.

A different meaning of the term hypothesis is used in formal logic to denote the antecedent of a proposition.

Thus,

In the proposition,

If P then Q,

P denotes the hypothesis or antecedent.

Q can be called a consequent.

P is the assumption in a possibly counterfactual what-if question.

The adjective hypothetical,

Meaning having the nature of a hypothesis,

Or being assumed to exist as an immediate consequence of a hypothesis,

Can refer to any of these meanings of the term hypothesis.

In its ancient usage,

Hypothesis referred to a summary of the plot of a classical drama.

The English word hypothesis comes from the ancient Greek word hypothesis,

Whose literal or etymological sense is putting or placing under,

And hence,

In extended use,

Has many other meanings,

Including supposition.

In Plato's Mino,

Socrates dissects virtue with a method used by mathematicians,

That of investigating from a hypothesis.

In this sense,

Hypothesis refers to a clever idea or to a convenient mathematical approach that simplifies cumbersome calculations.

Cardinal Bellarmine gave a famous example of this usage in the warning issued to Galileo in the early 17th century,

That he must not treat the motion of the earth as a reality,

But merely as a hypothesis.

In common usage in the 21st century,

A hypothesis refers to a provisional idea whose merit requires evaluation.

For proper evaluation,

The framer of a hypothesis needs to define specifics in operational terms.

A hypothesis requires more work than a hypothesis.

A hypothesis requires more work by the researcher in order to either confirm or disprove it.

In due course,

A confirmed hypothesis may become part of a theory,

Or occasionally it may grow to become a theory itself.

Normally,

Scientific hypotheses have the form of a mathematical model.

Sometimes,

But not always,

One can always formulate them as existential statements,

Stating that some particular instance of the phenomenon under examination 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.

In entrepreneurial settings,

A hypothesis is used to formulate provisional ideas about the attributes of products or business models.

The formulated hypothesis is then evaluated,

Where the hypothesis is proven to be either true or false through verifiability or falsifiability-oriented experiment.

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 may also invoke statistics and only talk about probabilities.

Karl Popper,

Following others,

Has argued that a hypothesis must be falsifiable and that one cannot regard a proposition or theory as scientific if it does not admit the possibility of being shown to be false.

Other philosophers of science have rejected the criterion of falsifiability or supplemented it with other criteria,

Such as verifiability,

E.

G.

Verificationism,

Or coherence,

E.

G.

Confirmation holism.

The scientific method involves experimentation to test the ability of some hypothesis to adequately answer the question under investigation.

In contrast,

Unfettered observation is not as likely to raise unexplained issues or open questions in science,

As would the formulation of a crucial experiment to test the hypothesis.

A thought experiment might also be used to test the hypothesis.

In framing a hypothesis,

The investigator must not currently know the outcome of a test or that it remains reasonably under continuing investigation.

Only in such cases does the experiment,

Test,

Or study potentially increase the probability of showing the truth of a hypothesis.

If the researcher already knows the outcome,

It counts as a consequence,

And the researcher should have already considered this while formulating the hypothesis.

If one cannot assess the predictions by observation or by experience,

The hypothesis needs to be tested by others providing observations.

For example,

A new technology or theory might make the necessary experiments feasible.

A trial solution to a problem is commonly referred to as a hypothesis,

Or often as an educated guess,

Because it provides a suggested outcome based on the evidence.

However,

Some scientists reject the term educated guess as incorrect.

Experimenters may test and reject several hypotheses before solving the problem.

According to Schnick and Vaughn,

Researchers weighing up alternative hypotheses may take into consideration testability,

Compare falsifiability as discussed above,

Parsimony,

As in the application of Occam's razor,

Discouraging the postulation of excessive numbers of entities,

Scope,

The apparent applicability of the hypothesis to multiple known phenomena,

Fruitfulness,

The prospect that the hypothesis may explain further phenomena in the future,

Conservatism,

The degree of fit with existing recognized knowledge systems.

A working hypothesis is a hypothesis that is provisionally accepted as a basis for further research in the hope that a tenable theory will be produced even if the hypothesis ultimately fails.

Like all hypotheses,

A working hypothesis is constructed as a statement of expectations,

Which can be linked to the exploratory research purpose in empirical investigation.

Working hypotheses are often used as a conceptual framework in qualitative research.

The provisional nature of working hypotheses makes them useful as an organizing device in applied research.

Here they act like a useful guide to address problems that are still in a formative phase.

In recent years,

Philosophers of science have tried to integrate the various approaches to evaluating hypotheses,

And the scientific method in general,

To form a more complete system that integrates the individual concerns of each approach.

Notably,

Imre Lakatos and Paul Fjärbend,

Karl Popper's colleagues and students,

Respectively,

Have produced novel attempts at such a synthesis.

Concepts in Hempel's deductive nomological model play a key role in the development and testing of hypotheses.

Most formal hypotheses connect concepts by specifying the expected relationships between propositions.

When a set of hypotheses are grouped together,

They become a type of conceptual framework.

When a conceptual framework is complex and incorporates causality or explanation,

It is generally referred to as a theory.

According to noted philosopher of science,

Carl Gustav Hempel,

Inadequate empirical interpretation turns a theoretical system into a testable theory.

The hypotheses,

Whose constituent terms have been interpreted,

Become capable of test by reference to observable phenomena.

Frequently,

The interpreted hypothesis will be derivative hypotheses of the theory,

But their confirmation or disconfirmation by empirical data will then immediately strengthen or weaken also the primitive hypotheses from which they were derived.

Hempel provides a useful metaphor that describes the relationship between a conceptual framework and the framework as it is observed and,

Perhaps,

Tested interpretive framework.

The whole system floats,

As it were,

Above the plane of observation and is anchored to it by rules of interpretation.

These might be viewed as strings which are not part of the network,

But link certain points of the latter with specific places in the plane of observation.

By virtue of those interpretive connections,

The network can function as a scientific theory.

Hypotheses with concepts anchored in the plane of observation are ready to be tested.

In actual scientific practice,

The process of framing a theoretical structure and of interpreting it are not always sharply separated,

Since the intended interpretation usually guides the construction of the theoretician.

It is,

However,

Possible and indeed desirable for the purpose of logical clarification to separate the two steps conceptually.

When a possible correlation or similar relation between phenomena is investigated,

Such as whether a proposed remedy is effective in treating a disease,

The hypothesis that a relation exists cannot be examined the same way one might examine a proposed new law of nature.

In such an investigation,

If the tested remedy shows no effect in a few cases,

These do not necessarily falsify the hypothesis.

Instead,

Statistical tests are used to determine how likely it is that the overall effect would be observed if the hypothesized relation does not exist.

If that likelihood is sufficiently small,

E.

G.

Less than 1%,

The existence of a relation may be assumed.

Otherwise,

Any observed effect may be due to pure chance.

In statistical hypothesis testing,

Two hypotheses are compared.

These are called the null hypothesis and the alternative hypothesis.

The null hypothesis is the hypothesis that states that there is no relation between the phenomena whose relation is under investigation,

Or at least not of the form given by the alternative hypothesis.

The alternative hypothesis,

As the name suggests,

Is the alternative to the null hypothesis.

It states that there is some kind of relation.

The alternative hypothesis may take several forms,

Depending on the nature of the hypothesized relation.

In particular,

It can be two-sided.

For example,

There is some effect in a yet unknown direction,

Or one-sided.

The direction of the hypothesized relation,

Positive or negative,

Is fixed in advance.

Conventional significance levels for testing hypotheses,

Acceptable probabilities of wrongly rejecting a true null hypothesis,

Are 0.

10,

0.

05,

And 0.

01.

The significance level for deciding whether the null hypothesis is rejected and the alternative hypothesis is accepted must be determined in advance,

Before the observations are collected or inspected.

If these criteria are determined later,

When the data to be tested are already known,

The test is invalid.

The above procedure is actually dependent on the number of the participants,

Units,

Or sample size that are included in the study.

For instance,

To avoid having the sample size be too small to reject a null hypothesis,

It is recommended that one specify a sufficient sample size from the beginning.

It is advisable to define a small,

Medium,

And large effect size for each of a number of important statistical tests,

Which are used to test the hypothesis.

Honors.

Mount Hypothesis in Antarctica is named in appreciation of the role of hypothesis in scientific research.

Several hypotheses have been put forth in different subject areas.

Astronomical hypotheses.

Authorship debates.

Biological hypotheses.

Documentary hypothesis.

Hypothetical documents.

Hypothetical impact events.

Hypothetical laws.

Linguistic theories and hypotheses.

Meteorological hypotheses.

Hypothetical objects.

Origin hypotheses of ethnic groups.

Hypothetical processes.

Hypothetical spacecraft.

Statistical hypothesis testing.

Hypothetical technology.

An axiom,

Postulate,

Or assumption is a statement that is taken to be true.

To serve as a premise or starting point for further reasoning and arguments.

The word comes from the ancient Greek word axioma,

Meaning that which is thought worthy or fit,

Or that which commends itself as evident.

The precise definition varies across fields of study.

In classical philosophy,

An axiom is a statement that is so evident or well-established that it is accepted without controversy or question.

In modern logic,

An axiom is a premise or starting point for reasoning.

In mathematics,

An axiom may be a logical axiom or a non-logical axiom.

Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form.

E.

G.

A and b implies a.

While non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory.

For example,

A plus zero equals a in integer arithmetic.

Non-logical axioms may also be called postulates or assumptions.

In most cases,

A non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory and might or might not be self-evident in nature.

E.

G.

The parallel postulate in Euclidean geometry.

To axiomatize a system of knowledge is to show that its claims can be derived from a small well-understood set of sentences,

The axioms,

And there are typically many ways to axiomatize a given mathematical domain.

Any axiom is a statement that serves as a starting point from which other statements are logically derived.

Whether it is meaningful,

And if so,

What it means,

For an axiom to be true is a subject of debate in the philosophy of mathematics.

The word axiom comes from the Greek word axioma,

A verbal noun from the word axioin,

Meaning to deem worthy,

But also to require,

Which in turn comes from axios,

Meaning being in balance,

And hence having the same value as worthy,

Proper.

Among the ancient Greek philosophers and mathematicians,

Axioms were taken to be immediately evident propositions,

Foundational and common to many fields of investigation,

And self-evidently true without any further argument or proof.

The root meaning of the word postulate is to demand.

For instance,

Euclid demands that one agree that some things can be done,

E.

G.

Any two points can be joined by a straight line.

Ancient geometers maintained some distinction between axioms and postulates.

While commenting on Euclid's books,

Proclus remarks that Geminus held that this fourth postulate should not be classed as a postulate,

But as an axiom,

Since it does not,

Like the first three postulates,

Assert the possibility of some construction,

But expresses an essential property.

Boethius translated postulate as petitio and called the axioms notionis communis,

But in later manuscripts this usage was not always strictly kept.

The logico-deductive method whereby conclusions,

New knowledge,

Follow from premises,

Old knowledge,

Through the application of sound arguments,

Syllogisms,

Rules of inference,

Was developed by the ancient Greeks and has become the core principle of modern mathematics.

Tautology is excluded.

Nothing can be deduced if nothing is assumed.

Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge.

They are accepted without demonstration.

All other assertions,

Theorems,

In the case of mathematics,

Must be proven with the aid of these basic assumptions.

However,

The interpretation of mathematical knowledge has changed from ancient times to the modern,

And consequently the terms axiom and postulate hold a slightly different meaning for the present-day mathematician than they did for Aristotle and Euclid.

The ancient Greeks considered geometry as just one of several sciences and held the theorems of geometry on par with scientific facts.

As such,

They developed and used the logico-deductive method as a means of avoiding error,

And for structuring and communicating knowledge.

Aristotle's posterior analytics is a definitive exposition of the classical view.

An axiom in classical terminology referred to as self-evident assumption common to many branches of science.

A good example would be the assertion that when an equal amount is taken from equals,

An equal amount results.

At the foundation of the various sciences lay certain additional hypotheses that were accepted without proof.

Such a hypothesis was termed a postulate.

While the axioms were common to many sciences,

The postulates of each particular science were different.

Their validity had to be established by means of real-world experience.

Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.

The classical approach is well illustrated by Euclid's Elements,

Where a list of postulates is given,

Commonsensical geometric facts drawn from our experience,

Followed by a list of common notions,

Very basic self-evident assertions.

Postulates 1.

It is possible to draw a straight line from any point to any other point.

2.

It is possible to extend a line segment continuously in both directions.

3.

It is possible to describe a circle with any center in any radius.

4.

It is true that all right angles are equal to one another.

5.

Parallel postulate.

It is true that if a straight line following on two straight lines make the interior angles on the same side less than two right angles,

The two straight lines,

If produced indefinitely,

Intersect on that side on which are the angles less than the two right angles.

Common Notions.

1.

Things which are equal to the same thing are also equal to one another.

2.

If equals are added to equals,

The wholes are equal.

3.

If equals are subtracted from equals,

The remainders are equal.

4.

Things which coincide with one another are equal to one another.

5.

The whole is greater than the part.

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions,

Axioms,

Postulates,

Propositions,

Theorems,

And definitions.

One must concede the need for primitive notions or undefined terms or concepts in any study.

Such abstraction or formalization makes mathematical knowledge more general,

Capable of multiple differing meanings,

And therefore useful in multiple contexts.

Alessandro Padoa,

Mario Pieri,

And Giuseppe Peano were pioneers in this movement.

Structuralist mathematics goes further and develops theories and axioms,

E.

G.

Field theory,

Group theory,

Topology,

Vector spaces,

Without any particular application in mind.

The distinction between an axiom and a postulate disappears.

The postulates of Euclid are profitably motivated by saying that they led to a great wealth of geometric facts.

The truth of these complicated facts rests on the acceptance of the basic hypothesis.

However,

By throwing out Euclid's fifth postulate,

One can get theories that have meaning in wider contexts,

E.

G.

Hyperbolic geometry.

As such,

One must simply be prepared to use labels as line and parallel with greater flexibility.

The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements,

And not as facts based on experience.

When mathematicians employ the field axioms,

The intentions are even more abstract.

The propositions of field theory do not concern any one particular application.

The mathematician now works in complete abstraction.

There are many examples of fields.

Field theory gives correct knowledge about them all.

It is not correct to say that axioms of field theory are propositions that are regarded as true without proof.

Rather,

The field axioms are a set of constraints.

If any given system of addition and multiplication satisfies these constraints,

Then one is in a position to instantly know a great deal of extra information about the system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects,

And mathematics itself can be regarded as a branch of logic.

Friege,

Russell,

Poincaré,

Hilbert,

And Gödel are some of the key figures in this development.

Another lesson learned in modern mathematics is to examine purported proofs,

Carefully,

For hidden assumptions.

In the modern understanding,

A set of axioms is any collection of formally stated assertions from which other formally stated assertions follow,

By the application of certain well-defined rules.

In this view,

Logic becomes just another formal system.

A set of axioms should be consistent.

It should be impossible to derive a contradiction from the axioms.

A set of axioms should also be non-redundant.

An assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics,

Perhaps all of mathematics,

Could be derived from a consistent collection of basic axioms.

An early success of the formalist program was Hilbert's formalization of Euclidean geometry and the related demonstration of the consistency of those axioms.

In a wider context,

There was an attempt to base all of mathematics on Cantor's set theory.

Here,

The emergence of Russell's paradox and similar antinomies of naive set theory raised the possibility that any such system could turn out to be inconsistent.

The formalist project suffered a setback a century ago when Gödel showed that it is possible for any sufficiently large set of axioms,

Peano's axioms for example,

To construct a statement whose truth is independent of that set of axioms.

As a corollary,

Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers,

An infinite but intuitively accessible formal system.

However,

At present there is no known way of demonstrating the consistency of the modern Zermelo-Fraenkel axioms for set theory.

Furthermore,

Using techniques of forcing,

Cohen,

Who can show that the continuum hypothesis,

Cantor,

Is independent of the Zermelo-Fraenkel axioms.

Thus,

Even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

Experimental sciences,

As opposed to mathematics and logic,

Also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties,

Either still general or much more specialized to a specific experimental context.

For instance,

Newton's laws in classical mechanics,

Maxwell's equations in classical electromagnetism,

Einstein's equation in general relativity,

Mendel's laws of genetics,

Darwin's natural selection law,

Etc.

These founding assertions are usually called principles or postulates so as to distinguish from mathematical axioms.

As a matter of fact,

The role of axioms in mathematics and postulates in experimental sciences is different.

In mathematics,

One neither proves nor disproves an axiom.

A set of mathematical axioms gives a set of rules that fix a conceptual realm in which the theorems logically follow.

In contrast,

In experimental sciences,

A set of postulates shall allow deducing results that match or do not match experimental results.

If postulates do not allow deducing experimental predictions,

They do not set a scientific conceptual framework and have to be completed or made more accurate.

If the postulates allow deducing predictions of experimental results,

The comparison with experiments allows falsifying falsified a theory that the postulates install.

A theory is considered valid as long as it has not been falsified.

Now,

The transition between the mathematical axioms and scientific postulates is always slightly blurred,

Especially in physics.

This is due to the heavy use of mathematical tools to support the physical theories.

For instance,

The introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply.

It became more apparent when Albert Einstein first introduced special relativity,

Where the invariant quantity is no more the Euclidean length but the Minkowski space-time interval,

And then general relativity where flat Minkowskian geometry is replaced with pseudo-Riemannian geometry on curved manifolds.

In quantum physics,

Two sets of postulates have coexisted for some time,

Which provide a very nice example of falsification.

The Copenhagen School,

Niels Bohr,

Werner Heisenberg,

Max Born,

Developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors,

States,

In separable Hilbert space,

And physical quantities as linear operators that act in this Hilbert space.

This approach is fully falsifiable and has so far produced the most accurate predictions in physics,

But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask.

For this reason,

Another hidden variables approach was developed for some time by Albert Einstein,

Erwin Schrödinger,

David Bohm.

It was created so as to try to give deterministic explanation to phenomena such as entanglement.

This approach assumed that the Copenhagen School description was not complete,

And postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer,

The founding elements of which were discussed as the EPR paradox in 1935.

Taking this idea seriously,

John Bell derived in 1964 a prediction that would lead to different experimental results,

Bell's inequalities,

In the Copenhagen and the hidden variable case.

The experiment was conducted first by Alain Aspect in the early 1980s,

And the result excluded the simple hidden variable approach.

Sophisticated hidden variables could still exist,

But their properties would still be more disturbing than the problems they try to solve.

This does not mean that the conceptual framework of quantum physics can be considered as complete now,

Since some open questions still exist.

The limit between the quantum and classical realms,

What happens during a quantum measurement,

What happens in a completely closed quantum system such as the universe itself,

Etc.

In the field of mathematical logic,

A clear distinction is made between two notions of axioms,

Logical and non-logical,

Somewhat similar to the ancient distinction between axioms and postulates respectively.

Logical axioms are certain formulas in a formal language that are universally valid,

That is,

Formulas that are satisfied by every assignment of values.

Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language.

In the case of predicate logic,

More logical axioms than that are required in order to prove logical truths that are not tautologies in the strict sense.

Reviews

The hypothesis of sleep therapy worked well to lull to sleep. Cheers Ben