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In the
philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
, a non-surveyable proof is a
mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
that is considered infeasible for a human mathematician to
verify CONFIG.SYS is the primary configuration file for the DOS and OS/2 operating systems. It is a special ASCII text file that contains user-accessible setup or configuration directives evaluated by the operating system's DOS BIOS (typically residi ...
and so of controversial validity. The term was coined by Thomas Tymoczko in 1979 in criticism of Kenneth Appel and Wolfgang Haken's
computer-assisted proof Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machine ...
of the
four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...
, and has since been applied to other arguments, mainly those with excessive case splitting and/or with portions dispatched by a difficult-to-verify computer program. Surveyability remains an important consideration in
computational mathematics Computational mathematics is the study of the interaction between mathematics and calculations done by a computer.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006. Retri ...
.


Tymoczko's argument

Tymoczko argued that three criteria determine whether an argument is a mathematical proof: * ''Convincingness'', which refers to the proof's ability to persuade a rational prover of its conclusion; * ''Surveyability'', which refers to the proof's accessibility for verification by members of the human mathematical community; and * ''Formalizability'', which refers to the proof's appealing to only logical relationships between concepts to substantiate its argument. In Tymoczko's view, the Appel–Haken proof failed the surveyability criterion by, he argued, substituting
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
for deduction: Without surveyability, a proof may serve its first purpose of convincing a reader of its result and yet fail at its second purpose of enlightening the reader as to why that result is true—it may play the role of an observation rather than of an argument. Bonnie Gold and Roger Simons. Proof and Other Dilemmas: Mathematics and Philosophy.Giandomenico Sica. Essays on the Foundations of Mathematics and Logic. Volume 1. This distinction is important because it means that non-surveyable proofs expose mathematics to a much higher potential for error. Especially in the case where non-surveyability is due to the use of a computer program (which may have bugs), most especially when that program is not published, convincingness may suffer as a result. As Tymoczko wrote:


Counterarguments to Tymoczko's claims of non-surveyability

Tymoczko's view is contested, however, by arguments that difficult-to-survey proofs are not necessarily as invalid as impossible-to-survey proofs. Paul Teller claimed that surveyability was a matter of degree and reader-dependent, not something a proof does or does not have. As proofs are not rejected when students have trouble understanding them, Teller argues, neither should proofs be rejected (though they may be criticized) simply because professional mathematicians find the argument hard to follow.Paul Teller. "Computer Proof". The Journal of Philosophy. 1980. (Teller disagreed with Tymoczko's assessment that " he Four-Color Theoremhas not been checked by mathematicians, step by step, as all other proofs have been checked. Indeed, it cannot be checked that way.") An argument along similar lines is that case splitting is an accepted proof method, and the Appel–Haken proof is only an extreme example of case splitting.


Countermeasures against non-surveyability

On the other hand, Tymoczko's point that proofs must be at least possible to survey and that errors in difficult-to-survey proofs are less likely to fall to scrutiny is generally not contested; instead methods have been suggested to improve surveyability, especially of computer-assisted proofs. Among early suggestions was that of parallelization: the verification task could be split across many readers, each of which could survey a portion of the proof. But modern practice, as made famous by Flyspeck, is to render the dubious portions of a proof in a restricted formalism and then verify them with a proof checker that is available itself for survey. Indeed, the Appel–Haken proof has been thus verified.Julie Rehmeyer. "How to (Really) Trust a Mathematical Proof". ScienceNews. https://www.sciencenews.org/article/how-really-trust-mathematical-proof. Retrieved 2008-11-14. Nonetheless, automated verification has yet to see widespread adoption.Freek Wiedijk
The QED Manifesto Revisited
2007


References

{{reflist Mathematical proofs Proof theory Automated theorem proving