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위키데이터

• ID : Q11538

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1. High among the notions that cause not a few students to wonder if perhaps math is not the subject for them, is mathematical proof.
2. Though it is the bedrock of professional pure mathematics, the concept of proof is barely touched on outside university mathematics departments.
3. For sure, I have never in my life seen a proof that truly fits the standard definition.
4. It’s not a bad approach if the goal is to give someone a general idea of what a proof is.
5. The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master.
6. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction.
7. We begin in this section by characterizing two somewhat extreme reactions to visual proofs, which we term the Baroque and Romantic reactions to visual proof.
8. One might rigidly insist, for example, that a proof must be expressed so as to explicitly draw logical connections between mathematical propositions.
9. Logic operates on the syntax of sentences, so without sentences there can be no proof write-up.
10. Something that doesn't address the general phenomenon can't be a proof that the phenomenon holds universally.
11. Although, at the time, some of his close collaborators said they found the proof to be correct, experts around the world struggled, often reluctantly, to slog through it, let alone verify it.
12. Classification of students’ proof schemes offers insights into students’ mathematical proving.
13. However, previous studies revealed little about finer progressions in students’ proving and barely examined their proof schemes involving counterexamples.
14. This study examined students’ proof schemes through their considerations of examples and counterexamples and reasoning in Proof Constructions.
15. The refined classifications revealed students’ progression through their sophisticated use of examples, counterexamples, and deductive inferences and that Proof Construction was knowledge-driven.
16. 29 , one of the oldest surviving fragments of Euclid's, a textbook used for millennia to teach proof-writing techniques.
17. A mathematical proof is an inferential argument for a mathematical statement , showing that the stated assumptions logically guarantee the conclusion.
18. Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases.
19. Purely formal proofs , written fully in symbolic language without the involvement of natural language, are considered in proof theory .
20. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools.
21. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning.
22. This article is concerned with shedding some light on Mathematics students' perceptions of mathematical proof, and how these perceptions may change over their first year at University.
23. "Proof" has been and remains one of the concepts which characterises mathematics.
24. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc.
25. What is a proof that it yields understanding (when it does)?
26. But if that is right then we need to reconsider the notions of formal reasoning and formal proof.
27. In giving an informal proof, we try to cover the essential, unfamiliar, unobvious steps and omit the trivial and routine inferences".
28. But what exactly is the relationship between the mathematician's reasoning and the formalized proof?
29. That’s one reason mathematicians strive for proof, not just evidence.
30. It’s proof that establishes mathematical truth.
31. The twin primes conjecture is one example where evidence, as much as proof, guides our mathematical thinking.
32. A proof by construction is just that, we want to prove something by showing how it can come to be.
33. This book describes the language used in a mathematical proof and also the different types of proofs used in math.
34. According to Hardy (1999, pp. 15-16), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof.
35. There is some debate among mathematicians as to just what constitutes a proof.
36. The four-color theorem is an example of this debate, since its "proof" relies on an exhaustive computer testing of many individual cases which cannot be verified "by hand.
37. A page of proof-related humor is maintained by Chalmers.
38. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty.
39. University of Maryland mathematicians Jacob Bedrossian, Samuel Punshon-Smith and Alex Blumenthal have developed the first rigorous mathematical proof explaining a fundamental law of turbulence.
40. "We believe our proof provides the foundation for understanding why Batchelor's law, a key law of turbulence, is true in a way that no theoretical physics work has done so far.
41. A mathematical proof of the law can be considered as an ideal consistency check.
42. "Sometimes the method of proof can be even more important than the proof itself.
43. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice.
44. This dramatic shift allows the constructive, elementary definition of the syntax of theories and, in particular, of the concept of proof in a formal theory.
45. Meaningful mathematics is to be based, Bernays demands, on primitive intuitive knowledge that includes, however, induction concerning natural numbers—both as a proof and definition principle.
46. Bernays concludes the outline by suggesting, “This would be followed by the development of proof theory”.
47. The third part of these lectures is entitled The grounding of the consistency of arithmetic by Hilbert’s new proof theory.
48. If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach.
49. A proof is defined as a derivation of one proposition from another.
50. A finest proof of this kind I discovered in a book by I. Stewart.
51. This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings.
52. Together, Karine Chemla and her ensemble of scholars successfully make the case for revising the nineteenth-century portrait of the history of mathematical proof that prevails even today.

메타데이터

위키데이터

• ID : Q11538

Spacy 패턴 목록

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