≹

isomorphismes:

It’s not always possible to say A ≻ B or A ≺ B. Sometimes

• neither A nor B is smaller. A≹B
• neither A nor B is more successful. A≹B
• neither A nor B is prettier. A≹B
• neither A nor B is smarter. A≹B
• you don’t love A any more or any less than you love B. ℒ(A)≹ℒ(B)
• neither A nor B is tastier. A≹B
• neither A nor B is closer. |A−x|≹|B−x|
• neither A nor B is more fair. A≹B
• neither A nor B is better. A≹B

I’ve argued this before using posets. And I intend to argue it further later, when I claim that the concept of Pareto superiority was a major step forward in ethics.

*[The concept of Pareto dominance allows you to make, at least in theory, a valid, fully general comparison between two states of the world. A≻B in full generality iff   aᵢ≻ bᵢ  ∀ aᵢ ∈ A and ∀ bᵢ ∈ B, by the individual standards of ∀ ≻ᵢ.]

For now, though, I’ll draw some examples of functionals that don’t beat one another. That is, ƒ≹g nor g≹ƒ. (You would probably assume ≹ is 2-symmetric but I’m just going to state it for clarity.)

In this drawing, green wins sometimes and purple wins other times. Is it more important to win the “righthand” cases or the “lefthand” cases? How much better for each scenario? (see integrating kernel) Is it better for the L&sub2; norm to be higher? Or just for the ∫ mass to be greater?

In this drawing, orange wins sometimes and blue wins other times. Is it more important to win the “interior” cases or the “extremal” cases? How much better for each scenario? (see kernel of integration)

How about a function that measures the desirability of a particular boyfriend / girlfriend in various scenarios. How about the function g measures boyfriend B in the various scenarios (domain) and the function ƒ measures boyfriend A in the various scenarios. By measures, I mean the function’s codomain is some kind of totally ordered set where it does make sense to talk about better ≻ and worse ≺.

• ƒ(at dinner) ≻ g(at dinner)
• ƒ(career) ≺ g(career)
• ƒ(in bed) ≫ g(in bed)
• ƒ(with your family) ≺ g(with your family)
• ƒ(at the beach) ≺ g(at the beach)
• …and so on…

So how do you decide whether A≺B or B≺A? Perhaps you have your own priorities sorted so well that you can apply a kernel. Or perhaps A≹B in the final analysis.

I could make a comparable list for

• comparing two houses or apartments (well, this one’s closer to the park, but that one has that cozy breakfast nook),
• comparing two societies (one where the top marginal tax rate is 41% and one where the top marginal tax rate is 40%),
• and on and on.

Sometimes it’s hard to compare. Sometimes — like which of your kids do you love the best — it’s impossible to compare.

(Source: isomorphismes)

1. tawnya-i5-paganini979 reblogged this from isomorphismes
2. coalitiong9p liked this
3. corbinofoz reblogged this from proofmathisbeautiful
4. epsilons reblogged this from proofmathisbeautiful
5. humbleantecedents reblogged this from proofmathisbeautiful
6. contodapalabra28 reblogged this from proofmathisbeautiful
7. tarabwie reblogged this from proofmathisbeautiful
8. cautionuseonlyasdirected reblogged this from proofmathisbeautiful
9. randomunsolve liked this
10. isometries reblogged this from proofmathisbeautiful
11. tster75 liked this
12. windycaves liked this
13. eggsandmilkgetouttahere reblogged this from isomorphismes
14. patellarreflex liked this
15. aragetolive reblogged this from proofmathisbeautiful
16. 87thsouth liked this
17. wiitns liked this
18. blindspots reblogged this from proofmathisbeautiful
19. minerva122 reblogged this from proofmathisbeautiful
20. twentyavailable liked this
21. macavitymadcap liked this
22. restlessninja liked this
23. whynotobssess reblogged this from proofmathisbeautiful
24. naturalnumber liked this
25. labisclosed liked this
26. lzimm reblogged this from isomorphismes
27. lzimm liked this
28. inordi liked this
29. videodroner liked this
30. goodframesdontsavebadpaintings reblogged this from isomorphismes
31. boom-bap liked this
32. poniesonfire liked this
33. photoshopfanatic liked this
34. yo-yoyosh liked this
35. amberjax liked this
36. stevenhle liked this
37. racionalidadlimitada liked this
38. dive-into-moss liked this
39. ouiououi liked this
40. heyheyjayjay reblogged this from proofmathisbeautiful
41. faintedtowers liked this
42. resong reblogged this from proofmathisbeautiful
43. showsomedelight liked this
44. ohheyitsrenee liked this
45. terrancisms liked this
46. abstractnerd reblogged this from isomorphismes
47. wolfbonerflow reblogged this from proofmathisbeautiful
48. isomorphismes posted this