# Get PDF Educational Research: Proofs, Arguments, and Other Reasonings: 4

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. 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 Zermeloâ€”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. 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.

Early pioneers of these methods intended the work ultimately to be embedded in a classical proof-theorem framework, e. 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. 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.

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.

## TI-AIE: Developing mathematical reasoning: mathematical proof: View as single page

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. A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States. 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". 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.

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.

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 assessment of likelihoods of hypotheses when new evidence or information is acquired. 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. 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 argument. Sometimes, the abbreviation "Q. This abbreviation stands for "Quod Erat Demonstrandum" , which is Latin for "that which was to be demonstrated". From Wikipedia, the free encyclopedia. See also: History of logic. Main article: Direct proof. Main article: Mathematical induction.

Main article: Contraposition. Main article: Proof by contradiction. Main article: Proof by construction.

## Andreas Stylianides

Main article: Proof by exhaustion. Main article: Probabilistic method. Main article: Combinatorial proof. Main article: Nonconstructive proof. Main article: Statistical proof. Main article: Computer-assisted proof. Main article: Experimental mathematics. Animated visual proof for the Pythagorean theorem by rearrangement. Main article: Elementary proof. Main articles: Inductive logic and Bayesian analysis. Main articles: Psychologism and Language of thought.

Main article: Q. Logic portal Mathematics portal. University of British Columbia. Retrieved September 26, 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. The Nuts and Bolts of Proofs. Academic Press, Discrete Mathematics with Proof. Definition 3. Cambridge University Press. The development of logic New ed.

Oxford University Press. January []. No work, except The Bible, has been more widely used See in particular p. Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. 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' Educational Studies in Mathematics.

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