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Kevin Knuth on Measuring

Kevin Knuth on Measuring. July 8, 2007 (20). Measuring. Kevin H. Knuth Department of Physics University at Albany. Familiarity Breeds the Illusion of Understanding Anonymous. We are all Familiar with Measuring. We Measure All Sorts of Things. Measuring Things We Don’t Understand.

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Kevin Knuth on Measuring

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  1. Kevin KnuthonMeasuring July 8, 2007 (20)

  2. Measuring Kevin H. Knuth Department of Physics University at Albany

  3. Familiarity Breeds the Illusion of Understanding • Anonymous MaxEnt 2007

  4. We are all Familiar with Measuring MaxEnt 2007

  5. We Measure All Sorts of Things MaxEnt 2007

  6. Measuring Things We Don’t Understand http://www.phy.duke.edu/research/photon/qelectron/proj/infv/images/einstein-nb-large.jpg http://superstruny.aspweb.cz/images/fyzika/quantum/quantum_split.jpg MaxEnt 2007

  7. But Not Everything MaxEnt 2007

  8. Unification of Ideas MaxEnt 2007

  9. Counting

  10. In the Beginning… MaxEnt 2007

  11. A Caveman had a Collection of Rocks MaxEnt 2007

  12. An Equivalence Class of Rocks If the Cavemanlikes, he can treat each rock as being equivalent to any other rock in his collection. He then has an Equivalence Class of Rocks MaxEnt 2007

  13. Comparing Collections of Rocks Two Cavemen might like to compare their collections. MaxEnt 2007

  14. One-to-One Correspondence If they treat the rocks as belonging to an equivalence class, they can attempt to make a one-to-one correspondence One Caveman has more rocks than the other since there is not a one-to-one mapping between the two sets of rocks MaxEnt 2007

  15. Why Carry Your Rocks Around? Every time they want to compare their collections, Thog has to carry his rocks over a big hill to meet Bok who carried his rocks across a raging river. There must be an easier way to make comparisons MaxEnt 2007

  16. Sticks? Thog has an idea, he suggests that for every rock, he should pick up a stick. Carrying sticks is easier! Bok thinks this over… MaxEnt 2007

  17. One Stick Instead of a lot of sticks, Bok suggests making a mark on a stick for each rock. MaxEnt 2007

  18. Natural Numbers Thog, who wants to impress his smart friend agrees and further suggests that they should make fancy marks rather than a lot of them MaxEnt 2007

  19. One-to-One Correspondence 1 2 Bok gets to work… 3 MaxEnt 2007

  20. At Their Next Meeting… The two gentlemen compare their markings and enjoy a relaxing afternoon free from carrying rocks. 3 2 Comparing their markings, they find that Bok has more rocks than Thog. MaxEnt 2007

  21. Ordering MaxEnt 2007

  22. Not All Rocks Are The Same Bok decides that some of his rocks are more impressive than others. One of them is HUGE!!! From this point on he no longer treats the rocks as belonging to an equivalence class MaxEnt 2007

  23. Lifting Rocks Bok finds it easy to order these rocks in terms of how heavy they are to lift. MaxEnt 2007

  24. Binary Ordering Relation Bok lifts one rock and compares its weight to another…one pair at a time. Rock 1 Rock 2 Rock 1 is no heavier than Rock 2 R1  R2 MaxEnt 2007

  25. Binary Ordering Relation Rock 2 Rock 3 Rock 2 is no heavier than Rock 3 R2  R3 MaxEnt 2007

  26. Transitivity R1  R2 and R2  R3 implies that Rock 1 Rock 3 Rock 1 is no heavier than Rock 3 R1  R3 MaxEnt 2007

  27. Diagram Representing the Ordering Heavier Bok uses his results to order his rocks MaxEnt 2007

  28. Ranking 3 Heavier Bok then realizes that he can use the same marks to compare one rock to another. 2 1 MaxEnt 2007

  29. Measuring MaxEnt 2007

  30. Rock Hunting Bok has become quite sophisticated, and now that his collection is finally in order, he decides to go hunting for another rock. MaxEnt 2007

  31. Sophisticated Rock Hunting MaxEnt 2007

  32. Ranking 3 Heavier Bok’s new rock messes up his ranking! 2 1 MaxEnt 2007

  33. Generalizing Ranking 3 Heavier Bok’s realizes that he can generalize his concept of heavier, to degrees of heaviness 2 1 MaxEnt 2007

  34. Measuring 4.2 LR 2.6 LR 1.8 LR 1 LR (Little Rock) Heavier He now measures his rocks against his standard MaxEnt 2007

  35. Measuring 2.3 Easy 1.5 Easy Lighter 1 Easy Meanwhile Thog, who has sharpened up a bit, came up with a similar methodology MaxEnt 2007

  36. Next Meeting 2.3 Easy 1.5 Easy 1 Easy Thog, who is proud of his accomplishments, brings his ideas to Bok along with a sheet of tree bark with numbers describing his collection. MaxEnt 2007

  37. Not Easy to Compare 4.2 LR 2.3 Easy 1.5 Easy Lighter 2.6 LR Heavier 1.8 LR 1 Easy 1 LR MaxEnt 2007

  38. An Agreement Thog likes how Bok has ordered his collection, as he too is proud of his biggest rock. To compromise, Bok agrees to use Thog’s smallest rock as a unit of weight. Thog brings it to Bok so he can begin comparing. MaxEnt 2007

  39. Not Easy to Compare 4.2 LR 2.3 Easy 1.5 Easy Lighter 2.6 LR Heavier 1.8 LR 1 Easy 1 LR MaxEnt 2007

  40. Adopt a Standard Ordering 4.2 LR 2.3 Rocks Heavier 2.6 LR Heavier 1.5 Rocks 1.8 LR 1.0 Rock 1.0 LR MaxEnt 2007

  41. Perform a Regraduation 2.1 Rocks 2.3 Rocks Heavier 1.3 Rocks Heavier 1.5 Rocks 0.9 Rock 1.0 Rock 0.5 Rock MaxEnt 2007

  42. On Their Next Meeting… Thog is pleased that his little rock is a unit of measure. Bok is surprised to find that Thog’s rock is actually heavier than his… by 0.2 Rocks! MaxEnt 2007

  43. An Idea… I was thinking… Maybe we could use these ideas to describe things other than rocks? I don’t see why not! We might be able to describe anything we can order. MaxEnt 2007

  44. A Set of Apples Here is a set of apples MaxEnt 2007

  45. Ordering Apples We can order them according to weight Heavier MaxEnt 2007

  46. Partially Ordered Set Apple 1 is no HEAVIER than Apple 2 We can order them according to weight Heavier A partially ordered set is a set along with a binary ordering relation Called a Poset for short MaxEnt 2007

  47. Posets of Apples Apple 1 is no SWEETER than Apple 2 We can order them according to sweetness Sweeter MaxEnt 2007

  48. Posets of Apples Apple 1 is no RIPER than Apple 2 We can order them according to how ripe they are More Ripe MaxEnt 2007

  49. Posets of Apples This configuration is called a CHAIN More Ripe MaxEnt 2007

  50. Isomorphisms 5 4 3 More Ripe Is Greater than or Equal to 2 1 16 8 4 Divides 2 1 MaxEnt 2007

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