What Are All the Ways to Do a Mathematical Proof

Concept in mathematics

P. Oxy. 29, ane of the oldest surviving fragments of Euclid'south Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Suggestion 5.^[ane]

A mathematical proof is an inferential argument for a mathematical argument, showing that the stated assumptions logically guarantee the decision. The argument may use other previously established statements, such as theorems; merely every proof tin, in principle, be constructed using only certain bones or original assumptions known as axioms,^[2] ^[iii] ^[4] along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or not-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the argument is true in all possible cases. A suggestion that has not been proved but is believed to be true is known as a theorize, or a hypothesis if frequently used every bit an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, forth with natural language which usually admits some ambivalence. In nigh mathematical literature, proofs are written in terms of rigorous breezy logic. Purely formal proofs, written fully in symbolic language without the involvement of tongue, are considered in proof theory. The distinction between formal and breezy proofs has led to much exam of current and historical mathematical exercise, quasi-empiricism in mathematics, and so-chosen folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

History and etymology [edit]

The word "proof" comes from the Latin probare (to test). Related mod words are English "probe", "probation", and "probability", Spanish probar (to smell or taste, or sometimes bear upon or test),^[5] Italian provare (to try), and German probieren (to effort). The legal term "probity" means potency or credibility, the power of testimony to prove facts when given by persons of reputation or status.^{[half dozen]}

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.^[seven] It is likely that the thought of demonstrating a conclusion start arose in connection with geometry, which originated in applied problems of country measurement.^[viii] The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements.^[9] Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known.

Mathematical proof was revolutionized past Euclid (300 BCE), who introduced the evident method still in use today. It starts with undefined terms and axioms, propositions concerning the undefined terms which are assumed to be self-obviously true (from Greek "axios", something worthy). From this footing, the method proves theorems using deductive logic. Euclid's volume, the Elements, was read past anyone who was considered educated in the Westward until the middle of the 20th century.^[10] In addition to theorems of geometry, such every bit the Pythagorean theorem, the Elements as well covers number theory, including a proof that the square root of ii is irrational and a proof that at that place are infinitely many prime numbers.

Farther advances also took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more full general proofs with no dependence on geometric intuition. In the 10th century CE, the Iraqi mathematician Al-Hashimi worked with numbers as such, chosen "lines" only not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers.^[11] An anterior proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to evidence the binomial theorem and backdrop of Pascal'due south triangle. Alhazen also developed the method of proof by contradiction, equally the first attempt at proving the Euclidean parallel postulate.^[12]

Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternating sets of axioms, for instance Axiomatic set theory and Not-Euclidean geometry.

Nature and purpose [edit]

As adept, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a argument. The standard of rigor is non absolute and has varied throughout history. A proof can exist presented differently depending on the intended audience. In guild to gain acceptance, a proof has to see communal standards of rigor; an statement considered vague or incomplete may be rejected.

The concept of proof is formalized in the field of mathematical logic.^[thirteen] A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to report. Indeed, the field of proof theory studies formal proofs and their properties, the most famous and surprising beingness that near all axiomatic systems tin generate certain undecidable statements not provable inside the arrangement.

The definition of a formal proof is intended to capture the concept of proofs equally written in the exercise of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. Yet, outside the field of automated proof assistants, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic–synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 "2 Dogmas of Empiricism" that such a distinction is untenable.^[14]

Proofs may exist admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs which he found to be particularly elegant equally coming from "The Book", a hypothetical tome containing the most cute method(due south) of proving each theorem. The book Proofs from THE Volume, published in 2003, is devoted to presenting 32 proofs its editors find specially pleasing.

Methods of proof [edit]

Direct proof [edit]

In direct proof, the conclusion is established by logically combining the axioms, definitions, and before theorems.^[xv] For example, direct proof can be used to prove that the sum of two fifty-fifty integers is always fifty-fifty:

Consider two even integers ten and y. Since they are fifty-fifty, they tin can be written every bit 10 = iia and y = 2b, respectively, for some integers a and b. Then the sum is 10 +y = 2a + twob = 2(a+b). Therefore 10+y has 2 as a cistron and, past definition, is even. Hence, the sum of any two even integers is even.

This proof uses the definition of even integers, the integer backdrop of closure under add-on and multiplication, and the distributive property.

Proof by mathematical induction [edit]

Despite its name, mathematical consecration is a method of deduction, not a class of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next example. Since in principle the induction rule can be applied repeatedly (starting from the proved base example), it follows that all (commonly infinitely many) cases are provable.^[xvi] This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent, which can be used, for instance, to prove the irrationality of the square root of 2.

A mutual application of proof by mathematical induction is to evidence that a belongings known to hold for 1 number holds for all natural numbers:^[17] Allow $N = {1, 2, iii, 4, ...$ } be the fix of natural numbers, and let $P (northward)$ exist a mathematical statement involving the natural number $due north$ belonging to $Due north$ such that

(i) $P (ane)$ is truthful, i.eastward., $P (north)$ is true for $n = 1$ .
(two) $P (n +1)$ is true whenever $P (n)$ is truthful, i.eastward., $P (n)$ is true implies that $P (n +one)$ is true.
Then $P (n)$ is true for all natural numbers $northward$ .

For example, nosotros can prove past induction that all positive integers of the form $2 n - 1$ are odd. Let $P (northward)$ represent " $2 n - 1$ is odd":

(i) For

n = i

2 due north - 1 = 2(1) - 1 = 1

, and

1

is odd, since information technology leaves a residue of

1

when divided past

2

. Thus

P (1)

is truthful.

(two) For whatever

n

, if

2 n - 1

is odd (

P (n)

), so

(ii northward - 1) + 2

must besides be odd, because adding

2

to an odd number results in an odd number. But

(2 n - 1) + 2 = ii n + i = 2(n +1) - i

, so

2(north +one) - 1

is odd (

P (n +1)

). So

P (northward)

implies

P (n +one)

Thus

two due north - 1

is odd, for all positive integers

northward

The shorter phrase "proof by induction" is ofttimes used instead of "proof past mathematical induction".^[18]

Proof past contraposition [edit]

Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p".

For instance, contraposition can exist used to establish that, given an integer $x$ , if $x^{2}$ is even, then $x$ is fifty-fifty:

Suppose

x

is non even. And then

x

is odd. The product of 2 odd numbers is odd, hence

x^{two}=x\cdot x

is odd. Thus

x^{2}

is not even. Thus, if

10^{2}

is fifty-fifty, the supposition must be faux, so

ten

has to be even.

Proof by contradiction [edit]

In proof past contradiction, as well known by the Latin phrase reductio advertising absurdum (by reduction to the cool), information technology is shown that if some argument is causeless true, a logical contradiction occurs, hence the statement must exist fake. A famous case involves the proof that ${\sqrt {two}}$ is an irrational number:

Suppose that

{\sqrt {two}}

were a rational number. And so it could be written in lowest terms every bit

{\sqrt {2}}={a \over b}

where a and b are non-zilch integers with no mutual factor. Thus,

b{\sqrt {2}}=a

. Squaring both sides yields twob ² = a ². Since 2 divides the expression on the left, 2 must likewise divide the equal expression on the right. That is, a ² is fifty-fifty, which implies that a must as well be even, as seen in the suggestion above (in #Proof by contraposition). So nosotros can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b ⁱⁱ = (2c)² = 4c ^two. Dividing both sides past ii yields b ² = 2c ⁱⁱ. Just then, by the aforementioned argument as before, 2 divides b ², so b must be fifty-fifty. Withal, if a and b are both fifty-fifty, they take ii as a common gene. This contradicts our previous statement that a and b have no common factor, so nosotros must conclude that

{\sqrt {two}}

is an irrational number.

To paraphrase: if one could write ${\sqrt {2}}$ as a fraction, this fraction could never be written in everyman terms, since ii could ever be factored from numerator and denominator.

Proof by construction [edit]

Proof by construction, or proof past example, is the structure of a concrete case with a holding to evidence that something having that belongings exists. Joseph Liouville, for instance, proved the existence of transcendental numbers past constructing an explicit case. It can also exist used to construct a counterexample to disprove a proposition that all elements have a certain holding.

Proof by exhaustion [edit]

In proof by burnout, the conclusion is established past dividing it into a finite number of cases and proving each 1 separately. The number of cases sometimes tin can get very big. For instance, the get-go proof of the four color theorem was a proof by burnout with 1,936 cases. This proof was controversial because the majority of the cases were checked past a computer plan, not by hand. The shortest known proof of the four color theorem as of 2011^[update] even so has over 600 cases.^[19]

Probabilistic proof [edit]

A probabilistic proof is 1 in which an example is shown to exist, with certainty, by using methods of probability theory. Probabilistic proof, like proof past construction, is ane of many ways to prove existence theorems.

In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that at that place is a non-null probability that a called candidate will accept the desired property. This does not specify which candidates take the holding, only the probability could not be positive without at to the lowest degree one.

A probabilistic proof is non to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work on the Collatz conjecture shows how far plausibility is from genuine proof. While almost mathematicians exercise not think that probabilistic prove for the properties of a given object counts every bit a genuine mathematical proof, a few mathematicians and philosophers have argued that at to the lowest degree some types of probabilistic testify (such as Rabin's probabilistic algorithm for testing primality) are as good every bit genuine mathematical proofs.^[20] ^[21]

Combinatorial proof [edit]

A combinatorial proof establishes the equivalence of dissimilar expressions past showing that they count the same object in unlike ways. Frequently a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides 2 dissimilar expressions for the size of a unmarried set, again showing that the two expressions are equal.

Nonconstructive proof [edit]

A nonconstructive proof establishes that a mathematical object with a certain holding exists—without explaining how such an object tin be found. Often, this takes the form of a proof past contradiction in which the nonexistence of the object is proved to be impossible. In dissimilarity, a constructive proof establishes that a particular object exists by providing a method of finding information technology. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that $a^{b}$ is a rational number:

Either

{\sqrt {2}}^{\sqrt {2}}

is a rational number and we are done (have

a=b={\sqrt {2}}

), or

{\sqrt {two}}^{\sqrt {2}}

is irrational so we can write

a={\sqrt {2}}^{\sqrt {ii}}

and

b={\sqrt {2}}

. This then gives

\left({\sqrt {2}}^{\sqrt {2}}\correct)^{\sqrt {2}}={\sqrt {two}}^{ii}=2

, which is thus a rational number of the form

a^{b}.

Statistical proofs in pure mathematics [edit]

The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such equally involving cryptography, chaotic serial, and probabilistic number theory or analytic number theory.^[22] ^[23] ^[24] It is less commonly used to refer to a mathematical proof in the co-operative of mathematics known every bit mathematical statistics. See also the "Statistical proof using data" section below.

Figurer-assisted proofs [edit]

Until the twentieth century it was assumed that any proof could, in principle, exist checked by a competent mathematician to confirm its validity.^[vii] However, computers are now used both to evidence theorems and to deport out calculations that are too long for any human or team of humans to check; the starting time proof of the iv colour theorem is an case of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time mistake in its calculations calls the validity of such reckoner-assisted proofs into question. In do, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating back-up and cocky-checks into calculations, and past developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, specially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved.

Undecidable statements [edit]

A statement that is neither provable nor disprovable from a prepare of axioms is called undecidable (from those axioms). I example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry.

Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC.

Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.

Heuristic mathematics and experimental mathematics [edit]

While early mathematicians such as Eudoxus of Cnidus did not utilize proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential role of mathematics.^[25] With the increment in computing ability in the 1960s, meaning work began to be done investigating mathematical objects outside of the proof-theorem framework,^[26] in experimental mathematics. Early pioneers of these methods intended the work ultimately to exist embedded in a classical proof-theorem framework, e.g. the early on evolution of fractal geometry,^[27] which was ultimately so embedded.

[edit]

Visual proof [edit]

Although not a formal proof, a visual sit-in of a mathematical theorem is sometimes called a "proof without words". The left-paw picture below is an instance of a historic visual proof of the Pythagorean theorem in the case of the (iii,4,v) triangle.

Visual proof for the (three,four,5) triangle as in the Zhoubi Suanjing 500–200 BCE.
Blithe visual proof for the Pythagorean theorem by rearrangement.
A 2d blithe proof of the Pythagorean theorem.

Some illusory visual proofs, such every bit the missing square puzzle, tin can be constructed in a manner which appear to evidence a supposed mathematical fact but only do so nether the presence of tiny errors (for instance, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated.

Elementary proof [edit]

An elementary proof is a proof which simply uses basic techniques. More specifically, the term is used in number theory to refer to proofs that brand no use of complex analysis. For some time it was thought that certain theorems, like the prime theorem, could only exist proved using "college" mathematics. However, over time, many of these results have been reproved using just simple techniques.

2-column proof [edit]

A ii-column proof published in 1913

A item way of organising a proof using two parallel columns is often used as a mathematical exercise in uncomplicated geometry classes in the Us.^[28] The proof is written as a series of lines in ii columns. In each line, the left-hand column contains a suggestion, while the right-paw cavalcade contains a brief explanation of how the corresponding proposition in the left-mitt column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-mitt column is typically headed "Statements" and the correct-mitt cavalcade is typically headed "Reasons".^[29]

Vernacular use of "mathematical proof" [edit]

The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when information used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), specially when used to argue from data.

Statistical proof using data [edit]

"Statistical proof" from data refers to the awarding of statistics, information assay, or Bayesian analysis to infer propositions regarding the probability of information. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empirical show from exterior mathematics to verify. In physics, in add-on to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze information in a particle physics experiment or observational study in physical cosmology. "Statistical proof" may too refer to raw data or a convincing diagram involving data, such as scatter plots, when the information or diagram is adequately disarming without further analysis.

Inductive logic proofs and Bayesian analysis [edit]

Proofs using anterior logic, while considered mathematical in nature, seek to found propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than full certainty. Anterior logic should not be confused with mathematical induction.

Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new prove or information is acquired.

Proofs as mental objects [edit]

Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibniz, Frege, and Carnap have variously criticized this view and attempted to develop a semantics for what they considered to be the language of thought, whereby standards of mathematical proof might be applied to empirical science.^{[ citation needed ]}

Influence of mathematical proof methods outside mathematics [edit]

Philosopher-mathematicians such as Spinoza have attempted to formulate philosophical arguments in an axiomatic fashion, whereby mathematical proof standards could be applied to argumentation in full general philosophy. Other mathematician-philosophers take tried to utilize standards of mathematical proof and reason, without empiricism, to make it at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descartes' cogito argument.

Catastrophe a proof [edit]

Sometimes, the abbreviation "Q.E.D." is written to indicate the terminate of a proof. This abbreviation stands for "quod erat demonstrandum", which is Latin for "that which was to be demonstrated". A more common alternative is to use a foursquare or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation.

Meet also [edit]

Automatic theorem proving
Invalid proof
List of incomplete proofs
List of long proofs
Listing of mathematical proofs
Nonconstructive proof
Proof by intimidation
Termination analysis
Thought experiment
What the Tortoise Said to Achilles

References [edit]

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^ Clapham, C. & Nicholson, J.N. The Concise Oxford Lexicon of Mathematics, Quaternary edition. A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set up of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained.
^ Cupillari, Antonella (2005) [2001]. The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs (Third ed.). Bookish Press. p. 3. ISBN978-0-12-088509-i.
^ Gossett, Eric (July 2009). Discrete Mathematics with Proof. John Wiley & Sons. p. 86. ISBN978-0470457931. Definition 3.1. Proof: An Breezy Definition
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^ Hacking, Ian (1984) [1975]. The Emergence of Probability: A Philosophical Study of Early Ideas most Probability, Induction and Statistical Inference. Cambridge Academy Printing. ISBN978-0-521-31803-seven.
^ ^a ^b The History and Concept of Mathematical Proof, Steven Grand. Krantz. 1. February five, 2007
^ Kneale, William; Kneale, Martha (May 1985) [1962]. The evolution of logic (New ed.). Oxford Academy Press. p. 3. ISBN978-0-xix-824773-nine.
^ Moutsios-Rentzos, Andreas; Spyrou, Panagiotis (February 2015). "The genesis of proof in aboriginal Greece The pedagogical implications of a Husserlian reading". Archive ouverte HAL . Retrieved October 20, 2019.
^ Eves, Howard W. (Jan 1990) [1962]. An Introduction to the History of Mathematics (Saunders Series) (sixth ed.). Brooks/Cole. p. 141. ISBN978-0030295584. No work, except The Bible, has been more widely used...
^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences, 500 (1): 253–77 [260], Bibcode:1987NYASA.500..253M, doi:10.1111/j.1749-6632.1987.tb37206.x, S2CID 121416910
^ Eder, Michelle (2000), Views of Euclid'due south Parallel Postulate in Ancient Hellenic republic and in Medieval Islam, Rutgers University, retrieved January 23, 2008
^ Buss, Samuel R. (1998), "An introduction to proof theory", in Buss, Samuel R. (ed.), Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, pp. 1–78, ISBN978-0-08-053318-6 . See in particular p. three: "The study of Proof Theory is traditionally motivated by the trouble of formalizing mathematical proofs; the original formulation of beginning-order logic past Frege [1879] was the first successful step in this direction."
^ Quine, Willard Van Orman (1961). "Two Dogmas of Empiricism" (PDF). Universität Zürich — Theologische Fakultät. p. 12. Retrieved Oct 20, 2019.
^ Cupillari, p. twenty.
^ Cupillari, p. 46.
^ Examples of simple proofs by mathematical consecration for all natural numbers
^ Proof by induction Archived Feb 18, 2012, at the Wayback Car, University of Warwick Glossary of Mathematical Terminology
^ Run across Four color theorem#Simplification and verification.
^ Davis, Philip J. (1972), "Fidelity in Mathematical Soapbox: Is One and One Actually Ii?" American Mathematical Monthly 79:252–63.
^ Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." Journal of Philosophy 94:165–86.
^ "in number theory and commutative algebra... in particular the statistical proof of the lemma." [i]
^ "Whether abiding π (i.e., pi) is normal is a confusing problem without whatever strict theoretical sit-in except for some statistical proof"" (Derogatory apply.)[two]
^ "these observations suggest a statistical proof of Goldbach'south conjecture with very chop-chop vanishing probability of failure for large E" [3]
^ Mumford, David B.; Series, Caroline; Wright, David (2002). Indra's Pearls: The Vision of Felix Klein. Cambridge University Printing. ISBN978-0-521-35253-6. What to practice with the pictures? Two thoughts surfaced: the beginning was that they were unpublishable in the standard way, there were no theorems simply very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, just theorems were coins of the realm and the conventions of that day dictated that journals only published theorems.
^ "A Notation on the History of Fractals". Archived from the original on Feb 15, 2009. Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the reply ahead of time.
^ Lesmoir-Gordon, Nigel (2000). Introducing Fractal Geometry . Icon Books. ISBN978-1-84046-123-vii. ...brought habitation once again to Benoit [Mandelbrot] that there was a 'mathematics of the eye', that visualization of a problem was as valid a method every bit any for finding a solution. Amazingly, he found himself lonely with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym 'Bourbaki'...
^ Herbst, Patricio Chiliad. (2002). "Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Cavalcade Proof in the Early Twentieth Century" (PDF). Educational Studies in Mathematics. 49 (iii): 283–312. doi:10.1023/A:1020264906740. hdl:2027.42/42653. S2CID 23084607.
^ Dr. Fisher Burns. "Introduction to the Two-Column Proof". onemathematicalcat.org . Retrieved Oct 15, 2009.

External links [edit]

Look up proof in Wiktionary, the free dictionary.

Media related to Mathematical proof at Wikimedia Commons
Proofs in Mathematics: Unproblematic, Charming and Beguiling
A lesson well-nigh proofs, in a class from Wikiversity

gowanbabream.blogspot.com

Source: https://en.wikipedia.org/wiki/Mathematical_proof