The Unique Infinity of the Denumerable Reals - PowerPoint PPT Presentation

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The Unique Infinity of the Denumerable Reals

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  1. The Unique Infinityof the Denumerable Reals Mathematics on the Edge of Quantum Reality

  2. Dr. Brian L. Crissey • Professor of Mathematics • North Greenville University, SC • Math/CS 1975 • Johns Hopkins

  3. My Path • Started with Math • Then Physics • Saw better opportunities in Computer Science • But CS changed too quickly • Math seemed stable • Or so I thought

  4. Simplification One of Mathematics’ Great Traditions 12 / 4 = 3 = 0

  5. Today’s Intent א0א1א2א3 … To Simplify Transfinite Mathematics Down to… { φ } … the empty set

  6. Chart of Numbers Potentially Infinite Precision Finite Precision REALS RATIONALS INTEGERS 21/6 irrationals 21

  7. Infinite Periodic Precision • Periodic Reals have infinitely long decimal expansions • Example (1/7)10 • 0.142857142857142857142857… • Where do they fit?

  8. Repeating Expansions 1/7 Potentially Infinite Precision Finite Precision REALS RATIONALS INTEGERS 21/6 irrationals 21

  9. Eliminating Infinite Periodic Precision • Change the base to the denominator • (1/7)10 = (0.1) 7 • Radix is a presentation issue, not a characteristic of the number itself.

  10. Revised Chart of Numbers (0.1)7 Potentially Infinite Precision Finite Precision REALS RATIONALS 21/6 INTEGERS irrationals 21

  11. Are Irrationals Even Real? Leopold Kronecker 1823 - 1891 • Georg Cantor’s Mentor • Strongly disputed Cantor’s inclusion of irrationals as real numbers • “My dear Lord God made all the integers. Everything else is the work of Man.”

  12. Irrationals Never Reach The Real Number Line

  13. What is a Real Number? Reals are those numbers intended for measuring. Solomon Feferman 1928 – present • Mathematician and philosopher at Stanford University • Author of • In the Light of Logic

  14. Influential Disciplinesin the 20th Century Computer Science Physics QuantumTheory Computability Has Math Integratedthe New Knowledge?

  15. Mathematical Mindsfrom the Last Century • Physics • Quantum Theory • And the Limits of Measurability • Computer Science • Computability • And Enumeration • Time to Upgrade? Alan Turing Max Planck

  16. From Quantum Physics Everything is energy Matter is perception of concentrated energy Particle detector limit Smallest “particle” “Particles” “Waves” Δ

  17. Quantum Geometry A Quantum point occupies a non-zero volume Many implications A quantum “point” “Particles” “Waves” Δ

  18. Natural Units Max Planck suggested the establishment of Δ • “units of length, mass, time, and temperature that would … necessarily retain their significance for all times and all cultures, • even extraterrestrial and extrahuman ones, and which may therefore be designated as natural units of measure.”

  19. Planck Precision Limits Quantum-scale granulation of reality Mass Length Time Area Volume Density Any measure Δ Δ

  20. Planck Infinitesimals  L = lpl = (hG/c 3)1/2 = 10-33 cm  m = mpl = (hc/G)1/2 = 10-5 g  t = tpl= (hG/c 5)1/2 = 10-43 s

  21. Abraham Robinson, Mathematician • 1918 – 1974 • developed nonstandard analysis • a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics.

  22. Smallest Measurable Length  South Carolina is to a Proton  As a Proton is to a Planck length 

  23. The Quantum Limit is the limit of measurability. It is the quantum limitof X in the differential quotient of Calculus.

  24. Limited Real Precision If real numbers are for measuring, And measuring precision is limited by quantum mechanics, Then measurable real numbers have limited precision.

  25. A Lower Limit to Measurable Precision L= 10-35 m The “infinitesimal”

  26. The Measurable Universe is Granular  V

  27. Implication 1

  28. An Old Paradox Revisited • 1.999… = 1 + 9 * .111… • 1.999… = 1+ 9 * 1/9 • 1.999… = 1 + 1 • So 1.999… = 2 • But at the quantum edge, • 2 – 1.999… = Δ≠ 0 • So 2 ≠ 1.999… 1.9999999999999999999999999999999999999999999999999999999

  29. Classical 2:1 Point Paradox • There are exactly as many points in a line segment of length 2 as there are in a line segment of length 1. 2 1

  30. Reality Math 2:1 Paradox Revisited • The ratio of Δ-infinitesimals in a line segment of length 4 to those in a line segment of length 2 is 2:1.

  31. Classical Point-Density Paradox • There are exactly as many points in a line segment of length 1 as there are on the entire real number line.

  32. Reality-Math Point-Density Resolved • Rounding b to the nearest Δ-integer shows that a:b is many-to-one, not 1-to-1

  33. Pythagorus • Good Old Pythagorus • c2 = a2 + b2 • True for all right triangles • then and now and forever • Maybe

  34. Pythagorean Failures • The hypotenuse of a quantum-scale isosceles right triangle, being aΔ – integer, cannot be irrational. • Three cases pertain.

  35. Quantum Pythagorus Case 1 • The hypotenuse is a truncatedΔ– integer in a discontinuous triangle. • 9-9-12.729… • 9-9-12

  36. Quantum Pythagorus Case 2 • The hypotenuse is a rounded-upΔ– integer in a continuous triangle with overlap. • 9-9-12.729… • 9-9-13

  37. Quantum Pythagorus Case 3 • The triangle is continuous, • But the longest side is no hypotenuse because the triangle is not exactly right-angled.

  38. Quantum Pythagorean Triples • 3-4-5 • 5-12-13 • Is there a minimal angle? • 7-24-25?

  39. Quantum Geometry is Different • A = ½ BH • H = 2A / B • A = 15 balls • B = 5 balls • But H ≠ 6balls

  40. Geometry at the Quantum Edge of Reality • Circles, when pressed against each other • Become hexagons

  41. There are Three Regular Tesselations of the Plane • Nature chooses the hexagon

  42. Natural Angles and Forms • 60º • Equilateral triangles • No right triangles at the quantum edge

  43. Quantum Angles • Straight lines intersect at fixed angles of 60º and 120º

  44. Quantum Hexagonal Grid • Cartesian coordinates can translate into quantum hexagon sites

  45. What is a Quantum Circle? • A quantum circle is a hexagon

  46. Quantum Circles • Not all circumferences exist • Not all diameters exist • Not all “points” are equidistant from the center

  47. Quantum Continuity • Face-sharing may define continuity at the quantum edge • But it fails as a function.

  48. Quantum Discontinuity • Greater slopes cause discontinuity at the quantum edge • Only linear functions are continuous at the quantum edge

  49. Integration is Discrete • Quantum Integration is discrete • The integral is a Δ-sum • Discontinuous functions are integrable.

  50. Quantum 3-D Structures • What models will be useful in examining geometry at the quantum edge?