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- A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem — originally proved using techniques of complex analysis — was once thought to be a deep result until elementary proofs were found.
en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
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A deep result — or a deep theorem — is a mathematical statement whose proofs require a fundamentally new way of thinking, or a set of methods and techniques that are far beyond the concepts needed to formulate the claim.
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Apr 26, 2018 · Definition of the Deep End: In hyperbolic geometry, for any angle $\angle ABC$, there are points $D$ between the rays $\overrightarrow {BA}$ and $\overrightarrow {BC}$ such that none of the straights through $D$ can cross both rays $\overrightarrow {BA}$ and $\overrightarrow {BC}$.
- Introduction
- Some Theores That Are Considered Deep
- Criteria For Depth
- Deep Theorems at Undergraduate Level
- Is There A Natural Hierarchy of Axiom Systems?
- A Minimal Example of ‘Provable Depth’
- Conclusion
Mathematicians often call theorems ‘deep’, but the term has never been precisely defined. It is not easy even to explain the intuition behind it, though it probably includes the idea of being hidden, buried, and hard to uncover. A deep result, or a deep connection, is uncovered only by digging through several layers of concepts and proofs. Viewed f...
Here are four theorems that are widely regarded as deep. The first is much earlier than the others, and by today's standards not as deep. But it was notable as probably the deepest theorem of its time, and the harbinger of analytic methods in number theory. In fact, a proof of Dirichlet's theorem notusing analysis (a so-called ‘elementary’ proof) w...
The above theorems are ‘historically deep’ in the sense that it took a long time to uncover them, and many other theorems had to be uncovered first. But how well do they match some of the other criteria that have been thought to indicate depth? They are certainly difficult, surprising (in what it took to prove them), and their proofs are explanator...
The following theorems are sometimes covered at undergraduate level, because they are fundamental, fruitful, or the answer to natural questions. But they are deep enough to cause problems for students and others; so they are often glossed over or mentioned without proof. Even for those who become mathematicians, I suspect that one or more of these ...
The systems PA, PA+, PA++, … just described form a hierarchy of axiom systems that prove deeper and deeper theorems in the provability sense. We could measure the depth of a theorem by the height of the system required to prove it. However, this is a very rough and unsatisfactory way to measure depth. It is rough because even the lowest system PA...
The latter is not deep with the ancient Greek concept of area, because the triangle is equidecomposablewith a rectangle with the same base length and half the height: That is, with finitely many straight cuts the triangle can be decomposed and reassembled to form the rectangle (Figure 5). On the other hand, Euclid needed infinitely many cuts to fin...
History offers ample evidence of the concept of depth in mathematics, in the form of theorems depending on large, multi-level structures of concepts and lemmas. The trouble is: logic has so far offered only slight evidence that this concept can be formalized. It is true that there are several theorems of logic that offer possibilitiesof proving cer...
- John Stillwell
- 2015
deep A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem — originally proved using techniques of complex analysis — was once thought to be a deep result until elementary proofs were found. [ 1 ]
Length, width, height, and depth are nouns are derived from the adjectives long, wide, high, and deep. They follow a common English pattern that involves a vowel change (often to a shorter vowel) and the addition of th. (The lone t in height is modern.
May 2, 2024 · Use this glossary of over 150 math definitions for common and important terms frequently encountered in arithmetic, geometry, and statistics.
This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria.
