"গাণিতিক প্রমাণ" পাতাটির দুইটি সংশোধিত সংস্করণের মধ্যে পার্থক্য

ইংরেজি লেখা অপসারণ করা হলো
(১টি উৎস উদ্ধার করা হল ও ০টি অকার্যকর হিসেবে চিহ্নিত করা হল।) #IABot (v2.0.1)
(ইংরেজি লেখা অপসারণ করা হলো)
{{কাজ চলছে/২০১৯}}
[[চিত্র:Oxyrhynchus_papyrus_with_Euclid's_Elements.jpg|ডান|থাম্ব|250x250পিক্সেল|P. Oxy. 29, [[ইউক্লিড|ইউক্লিডের]] ''[[ইউক্লিডের উপাদানসমূহ|উপাদানসমূহ]]'' বইয়ের পাতার একটি ছেঁড়া অংশ।<ref>{{ওয়েব উদ্ধৃতি|ইউআরএল=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html|শিরোনাম=One of the Oldest Extant Diagrams from Euclid|লেখক=[[Bill Casselman (mathematician)|Bill Casselman]]|লেখক-সংযোগ=|তারিখ=|প্রকাশক=University of British Columbia|সংগ্রহের-তারিখ=September 26, 2008}}</ref>]]
'''গাণিতিক প্রমাণ''' হল [[গাণিতিক বিবৃতি]]<nowiki/>র জন্য এক ধরনের [[অনুমিতি]]<nowiki/>ক যুক্তি। এ ধরনের যুক্তিতে ইতোমধ্যে প্রতিষ্ঠিত বিভিন্ন বিবৃতি (যেমন- [[উপপাদ্য]]) ব্যবহার করা যায়। তাত্ত্বিকভাবে, অনুমিতির বিভিন্ন স্বীকৃত নিয়মের পাশাপাশি বেশকিছু অনুমিত বিবৃতির উপর নির্ভর করে একটি প্রমাণ সম্পন্ন করা হয়। এ ধরনের অনুমিত বিবৃতিগুলোকে গাণিতিক ভাষায় [[স্বতঃসিদ্ধ]] বলা হয়।<ref>{{বই উদ্ধৃতি|লেখক১=Clapham, C.|লেখক২=Nicholson, JN.|lastauthoramp=yes|শিরোনাম=The Concise Oxford Dictionary of Mathematics, Fourth edition|উক্তি=A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained.}}</ref><ref name="nutsandbolts">Cupillari, Antonella. ''The Nuts and Bolts of Proofs''. Academic Press, 2001. p. 3.</ref><ref>{{বই উদ্ধৃতি||শিরোনাম=Discrete Mathematics with Proof|তারিখ=July 2009|প্রথমাংশ=Eric|শেষাংশ=Gossett|পাতা=86|উক্তি=Definition 3.1. Proof: An Informal Definition|প্রকাশক=[[Wiley (publisher)|John Wiley & Sons]]|আইএসবিএন=978-0470457931}}</ref> স্বতঃসিদ্ধগুলোকে উক্ত বিবৃতি প্রমাণের ক্ষেত্রে প্রধান শর্ত হিসেবে ভাবা যেতে পারে। অর্থাৎ, একটি গাণিতিক বিবৃতি তখনই প্রমাণ করার যোগ্য হবে যখন উক্ত স্বতঃসিদ্ধগুলো উপস্থিত থাকবে। অনেকগুলো সমর্থনসূচক ঘটনা দেখিয়ে কোনো বিবৃতি গাণিতিকভাবে প্রমাণ করা যায় না। গাণিতিক প্রমাণের ক্ষেত্রে অবশ্যই দেখাতে হবে উক্ত বিবৃতিটি সব সময়ের জন্য সত্য (এক্ষেত্রে ''অনেক''গুলো ঘটনার পরিবর্তে ''সব''গুলো ঘটনা নিয়ে পর্যালোচনা করা যেতে পারে। সেক্ষেত্রে দেখাতে হবে যে উক্ত বিবৃতিতে ''সকল'' ঘটনার জন্য সত্য)। গাণিতিকভাবে প্রমাণিত নয়, কিন্তু সত্য হিসেবে ধরে নেয়া হয় — এমন বিবৃতিকে [[অনুমান]] বলা হয়।
 
গণিতের আরো অগ্রগতি হয় মধ্যযুগের [[ইসলাম]]ি গণিতের মাধ্যমে। গোড়ার দিকের গ্রিক প্রমাণাদি জ্যামিতিক প্রদর্শনের ওপর অতি নির্ভরশীল ছিল, কিন্তু মুসলিম গণিতবিদদের দ্বারা বিকশিত [[পাটিগণিত]] ও [[বীজগণিত]]ের ফলে সেই নির্ভরশীলতা আর থাকল না।
In the 10th century CE, the [[Iraqi people|Iraqi]] mathematician [[Al-Hashimi]] provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for "lines." He used this method to provide a proof of the existence of [[irrational number]]s.<ref>{{citation|last=Matvievskaya|first=Galina|year=1987|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]]|volume=500|issue=1|pages=253–77 [260]|doi=10.1111/j.1749-6632.1987.tb37206.x|bibcode=1987NYASA.500..253M}}</ref> An [[Mathematical induction|inductive proof]] for [[Arithmetic progression|arithmetic sequences]] was introduced in the ''Al-Fakhri'' (1000) by [[Al-Karaji]], who used it to prove the [[binomial theorem]] and properties of [[Pascal's triangle]]. Alhazen also developed the method of [[proof by contradiction]], as the first attempt at proving the [[Euclidean geometry|Euclidean]] [[parallel postulate]].<ref>{{Citation|last=Eder|first=Michelle|year=2000|title=Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam|url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html|publisher=[[Rutgers University]]|accessdate=January 23, 2008}}</ref>
 
Modern [[proof theory]] treats proofs as inductively defined [[data structure]]s. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see [[Axiomatic set theory]] and [[Non-Euclidean geometry]] for examples).
 
== প্রকৃতি ও উদ্দেশ্য ==
 
As practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; an [[argument]] considered vague or incomplete may be rejected.
 
The concept of a proof is formalized in the field of mathematical logic.<ref>{{citation|title=Handbook of Proof Theory|volume=137|series=Studies in Logic and the Foundations of Mathematics|editor-first=Samuel R.|editor-last=Buss|editor-link=Samuel Buss|publisher=Elsevier|year=1998|isbn=978-0-08-053318-6|contribution=An introduction to proof theory|pages=1–78|first=Samuel R.|last=Buss|authorlink=Samuel Buss}}. See in particular [https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA3 p.&nbsp;3]: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege [1879] was the first successful step in this direction."</ref> A [[formal proof]] is written in a [[formal language]] instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas. Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of [[proof theory]] studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show that certain [[Independence (mathematical logic)|undecidable statements]] are not provable.
 
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated [[proof assistant]]s, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are [[Analytic proposition|analytic]] or [[Synthetic proposition|synthetic]]. [[Immanuel Kant|Kant]], who introduced the [[analytic–synthetic distinction]], believed mathematical proofs are synthetic.
 
Proofs may be viewed as aesthetic objects, admired for their [[mathematical beauty]]. The mathematician [[Paul Erdős]] was known for describing proofs he found particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book ''[[Proofs from THE BOOK]]'', published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
 
== পদ্ধতিসমূহ ==
=== গাণিতিক আরোহ পদ্ধতিতে প্রমাণ ===
{{মূল নিবন্ধ|গাণিতিক আরোহ পদ্ধতি}}
Despite its name, mathematical induction is a method of [[Deductive reasoning|deduction]], not a form 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 [[Material conditional|implies]] the next case. Since in principle the induction rule can be applied repeatedly starting from the proved base case, we see that all (usually [[Infinite set|infinitely]] many) cases are provable.<ref>Cupillari, p. 46.</ref> This avoids having to prove each case individually. A variant of mathematical induction is [[proof by infinite descent]], which can be used, for example, to prove the [[Square root of 2#Proofs of irrationality|irrationality of the square root of two]].
 
গাণিতিক আরোহ পদ্ধতি হ্রাস করার একটি পদ্ধতি, এটি আরোহী যুক্তির কোনও রূপ নয়।
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers:<ref>[http://zimmer.csufresno.edu/~larryc/proofs/proofs.mathinduction.html Examples of simple proofs by mathematical induction for all natural numbers]</ref>
 
গাণিতিক আরোহ পদ্ধতিতে প্রমাণের একটি সাধারণ উদাহরণ হলো একটি সংখ্যার জন্য প্রযোজ্য বৈশিষ্ট্য সকল প্রাকৃতিক সংখ্যার জন্য প্রযোজ্য:<ref>[http://zimmer.csufresno.edu/~larryc/proofs/proofs.mathinduction.html Examples of simple proofs by mathematical induction for all natural numbers]</ref>
 
মনে করি, {{math|1='''N''' = {1,2,3,4,...}}} একটি স্বাভাবিক সংখ্যা সেট, এবং {{math|''P''(''n'')}} একটি গাণিতিক বিবৃতি যেখানে {{math|''n''}} একটি স্বাভাবিক সংখ্যা যা {{math|'''N'''}} এর অন্তর্গত যেন
 
: মনে করি, <math>\sqrt{2}</math> একটি [[মূলদ সংখ্যা]]। সংজ্ঞানুসারে, <math>\sqrt{2} = {a\over b}</math> যেখানে ''a'' এবং ''b'' শুন্য নয় এমন দুটি [[পূর্ণ সংখ্যা|পূর্ণসংখ্যা]] ও [[সহমৌলিক]]। ফলে, <math>b\sqrt{2} = a</math>। এর উভয়দিকে বর্গ করে পাই, 2''b''<sup>2</sup> = ''a''<sup>2</sup> । যেহেতু সমীকরণটির বামপক্ষ 2 দ্বারা বিভাজ্য, সেহেতু এর ডানপক্ষও 2 দ্বারা বিভাজ্য হবে (অন্যথায় একটি জোড় ও একটি বিজোড় সংখ্যা পরস্পর সমান হবে, যা অসম্ভব)। সুতরাং ''a''<sup>2</sup> জোড় সংখ্যা। অর্থাৎ ''a-''ও একটি জোড় সংখ্যা কেননা জোড় সংখ্যা বর্গ সর্বদা জোড় এবং বিজোড় সংখ্যার বর্গ সর্বদা বিজোড়। অতএব আমরা লিখতে পারি ''a'' = 2''c'', যেখানে ''c'' একটি পূর্ণসংখ্যা। একে মূল সমীকরণে [[প্রতিস্থাপন প্রক্রিয়া (গণিত)|প্রতিস্থাপন]] করে পাই 2''b''<sup>2</sup> = (2''c'')<sup>2</sup> = 4''c''<sup>2</sup>। প্রাপ্ত সমীকরণের উভয়দিকে 2 দ্বারা ভাগ করে পাই ''b''<sup>2</sup> = 2''c''<sup>2</sup>। পূর্বের যুক্তি অনুসারে বলা যায় যে ''b''<sup>2</sup>-ও 2 দ্বারা বিভাজ্য। অর্থাৎ ''b-''ও একটি জোড় সংখ্যা। কিন্তু যদি ''a'' এবং ''b'' উভয়ই জোড় হয়, তাহলে তাদের মধ্যে অবশ্যই একটা সাধারণ [[গুণিতক]] (এক্ষেত্রে 2) থাকবে যা শুরুতে করা অনুমান (অর্থাৎ ''a'' এবং ''b'' যে সহমৌলিক এই অনুমান)-এর সাথে অসঙ্গতি প্রকাশ করে। সুতরাং আমরা এ উপসংহারে পৌঁছুতে পারি যে <math>\sqrt{2}</math> একটি অমূলদ সংখ্যা।
 
=== গঠন দ্বারা প্রমাণ ===
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. [[Joseph Liouville]], for instance, proved the existence of [[transcendental number]]s by constructing an [[Liouville number|explicit example]]. It can also be used to construct a [[counterexample]] to disprove a proposition that all elements have a certain property.
 
=== নির্জীব প্রমাণ ===
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the [[four color theorem]] was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four color theorem {{As of|2011|lc=on}} still has over 600 cases.
 
=== সম্ভাব্যতার দ্বারা প্রমাণ ===
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of [[probability theory]]. Probabilistic proof, like proof by construction, is one of many ways to show [[existence theorem]]s.
 
This is not 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 most mathematicians do not think that probabilistic evidence ever counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's [[probabilistic algorithm]] for testing primality) are as good as genuine mathematical proofs.<ref>Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" ''American Mathematical Monthly'' 79:252–63.</ref><ref>Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." ''Journal of Philosophy'' 94:165–86.</ref>
 
=== কম্বিনেটরিক্সের মাধ্যমে প্রমাণ ===
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a [[Bijective proof|bijection]] between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a [[Double counting (proof technique)|double counting argument]] provides two different expressions for the size of a single set, again showing that the two expressions are equal.
 
=== Nonconstructive proof ===
A nonconstructive proof establishes that a [[mathematical object]] with a certain property exists without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a
nonconstructive proof shows that there exist two [[irrational number]]s ''a'' and ''b'' such that <math>a^b</math> is a [[rational number]]:
 
: Either <math>\sqrt{2}^{\sqrt{2}}</math> is a rational number and we are done (take <math>a=b=\sqrt{2}</math>), or <math>\sqrt{2}^{\sqrt{2}}</math> is irrational so we can write <math>a=\sqrt{2}^{\sqrt{2}}</math> and <math>b=\sqrt{2}</math>. This then gives <math>\left (\sqrt{2}^{\sqrt{2}}\right )^{\sqrt{2}}=\sqrt{2}^{2}=2</math>, which is thus a rational of the form <math>a^b.</math>
 
=== Statistical proofs in pure mathematics ===
The expression "statistical proof" may be used technically or colloquially in areas of [[pure mathematics]], such as involving [[cryptography]], [[chaotic series]], and probabilistic or analytic [[number theory]].<ref>"in number theory and commutative algebra... in particular the statistical proof of the lemma." [https://www.jstor.org/pss/2686395]</ref><ref>"Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some ''statistical'' proof"" (Derogatory use.)[http://www.springerlink.com/content/nj34v59p71m11125/]{{অকার্যকর সংযোগ|তারিখ=ফেব্রুয়ারি ২০২০ |bot=InternetArchiveBot |ঠিক করার প্রচেষ্টা=yes }}</ref><ref>"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E" [http://people.web.psi.ch/gassmann/eneseminare/abstracts/Goldbach1.pdf]</ref> It is less commonly used to refer to a mathematical proof in the branch of mathematics known as [[mathematical statistics]]. See also "[[গাণিতিক প্রমাণ#Colloquial use, Statistical proof using data|Statistical proof using data]]" section below.
 
=== Computer-assisted proofs ===
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.<ref name="Krantz">[http://www.math.wustl.edu/~sk/eolss.pdf The History and Concept of Mathematical Proof], Steven G. Krantz. 1. February 5, 2007</ref> However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the [[four color theorem]] is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight.
 
== অনিষ্পন্ন বিবৃতি ==
A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One 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 incompleteness theorem|Gödel's (first) incompleteness theorem]] shows that many axiom systems of mathematical interest will have undecidable statements.
 
== Heuristic mathematics and experimental mathematics ==
While early mathematicians such as [[Eudoxus of Cnidus]] did not use proofs, from [[Euclid]] to the [[foundational mathematics]] developments of the late 19th and 20th centuries, proofs were an essential part of mathematics.<ref>"''What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm and the conventions of that day dictated that journals only published theorems''", [[David Mumford]], Caroline Series and David Wright, [[Indra's Pearls (book)|Indra's Pearls]], 2002</ref> With the increase in computing power in the 1960s, significant work began to be done investigating [[mathematical objects]] outside of the proof-theorem framework,<ref>"''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 answer ahead of time.''"[http://home.att.net/~fractalia/history.htm A Note on the History of Fractals] {{ওয়েব আর্কাইভ|ইউআরএল=https://web.archive.org/web/20090215114618/http://home.att.net/~fractalia/history.htm |তারিখ=১৫ ফেব্রুয়ারি ২০০৯ }},</ref> in [[experimental mathematics]]. Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e.g. the early development of [[fractal geometry]],<ref>"''... brought home again to Benoit [Mandelbrot] that there was a 'mathematics of the eye', that visualization of a problem was as valid a method as any for finding a solution. Amazingly, he found himself alone with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym 'Bourbaki'... ''", Introducing Fractal Geometry, Nigel Lesmoir-Gordon</ref> which was ultimately so embedded.
 
== সম্পর্কিত বিষয়সমূহ ==
 
=== Visual proof ===
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "[[proof without words]]". The left-hand picture below is an example of a historic visual proof of the [[Pythagorean theorem]] in the case of the (3,4,5) triangle.
<gallery>
Image:Chinese pythagoras.jpg|Visual proof for the (3, 4, 5) triangle as in the [[Zhoubi Suanjing]] 500–200&nbsp;BCE.
File:Pythagoras-2a.gif|Animated visual proof for the Pythagorean theorem by rearrangement.
File:Pythag anim.gif|A second animated proof of the Pythagorean theorem.
</gallery>
 
Some illusory visual proofs, such as the [[missing square puzzle]], can be constructed in a way which appear to prove a supposed mathematical fact but only do so under the presence of tiny errors (for example, 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 ===
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in [[number theory]] to refer to proofs that make no use of [[complex analysis]]. For some time it was thought that certain theorems, like the [[prime number theorem]], could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
 
=== Two-column proof ===
[[চিত্র:Twocolumnproof.png|ডান|থাম্ব|A two-column proof published in 1913]]
 
A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States.<ref>{{সাময়িকী উদ্ধৃতি|প্রথমাংশ=Patricio G.|শেষাংশ=Herbst|শিরোনাম=Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century|সাময়িকী=[[Educational Studies in Mathematics]]|খণ্ড=49|সংখ্যা নং=3|বছর=2002|পাতাসমূহ=283–312|ডিওআই=10.1023/A:1020264906740}}</ref> The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons".<ref>[http://www.onemathematicalcat.org/Math/Geometry_obj/two_column_proof.htm Introduction to the Two-Column Proof], Carol Fisher</ref>
 
=== Colloquial use of "mathematical proof" ===
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 data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from [[data]].
 
=== Statistical proof using data ===
"Statistical proof" from data refers to the application of [[statistics]], [[data analysis]], or [[Bayesian analysis]] to infer propositions regarding the [[probability]] of [[data]]. 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 evidence from outside mathematics to verify. In [[physics]], in addition to statistical methods, "statistical proof" can refer to the specialized ''[[mathematical methods of physics]]'' applied to analyze data in a [[particle physics]] [[experiment]] or [[observational study]] in [[physical cosmology]]. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as [[scatter plot]]s, when the data or diagram is adequately convincing without further analysis.
 
=== Inductive logic proofs and Bayesian analysis ===
Proofs using [[inductive logic]], while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to [[probability]], and may be less than full [[certainty]]. Inductive logic should not be confused with [[mathematical induction]].
 
Bayesian analysis uses [[Bayes' theorem]] to update a person's [[Bayesian probability|assessment of likelihoods]] of hypotheses when new [[evidence]] or [[information]] is acquired.
 
=== Proofs as mental objects ===
Psychologism views mathematical proofs as psychological or mental objects. Mathematician [[philosopher]]s, such as [[Gottfried Wilhelm Leibniz|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|date=November 2014}}
 
=== Influence of mathematical proof methods outside mathematics ===
Philosopher-mathematicians such as [[Spinoza]] have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the [[certainty]] of propositions deduced in a mathematical proof, such as [[Descartes]]' [[Cogito ergo sum|''cogito'']] argument.
 
== প্রমাণের সমাপ্তিতে ==