Editing Mathematical proof (section)

==Related concepts==

===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>
File: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 by neglecting 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===
{{Main|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===
[[File:twocolumnproof.png|thumb|right|A two-column proof published in 1913]]

A particular way of organising a proof using two parallel columns is often used as a [[mathematical exercise]] in elementary geometry classes in the United States.<ref>{{cite journal |first=Patricio G. |last=Herbst |title=Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century |journal=[[Educational Studies in Mathematics]] |volume=49 |issue=3 |year=2002 |pages=283–312 |doi=10.1023/A:1020264906740 |hdl=2027.42/42653 |s2cid=23084607 |url=https://deepblue.lib.umich.edu/bitstream/2027.42/42653/1/10649_2004_Article_5096042.pdf |hdl-access=free }}</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>{{cite web|url=https://www.onemathematicalcat.org/Math/Geometry_obj/two_column_proof.htm |title=Introduction to the Two-Column Proof |author=Dr. Fisher Burns |website=onemathematicalcat.org |access-date=October 15, 2009}}</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 object]]s, 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===
{{Main|Statistical proof}}
"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===
{{Main|Inductive logic|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===
{{Main|Psychologism|Language of thought}}
Psychologism views mathematical proofs as psychological or mental objects.  Mathematician philosophers, such as [[Gottfried Wilhelm Leibniz|Leibniz]], [[Gottlob Frege|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]].

===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.