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Posts tagged with implication

Einstein opined that the great philosophical breakthrough leading to the mental possibility of science was the hypothetico-deductive method.

Which is a jargony way of saying: forget whether A is true or not (measurement of the world)—let’s talk about the separate, purely logical issue, of whetherif A were true, would B necessarily be true as well, as a result of A being true? ⧝

People aren’t great with hypotheticals, though—at least not everyone or not without education.

  • I can get people to agree with my reasoning  by first telling them that  leads to a conclusion they already agree with B.
  • (This is really dastardly because once I’ve judoed someone this far I can get them to agree to even more things, in order to maintain local consistency.)
  • We judge each other on credentials (A).
  • We judge arguments on what other experts think of them.
  • Mathematics is all about the  and most people are either scared to tears by mathematics, bored to tears by mathematics, or think mathematics irrelevant, or all three.
  • People think that if I argue that their reasoning  is wrong, I’m saying their conclusion B is wrong.
  • (Symbolically it’s obvious that A↛B = A⊬B = ¬(A→B) isn’t the same as ¬B. But people regularly interpret “That does not follow” as “That’s wrong”.)

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Whatever it is people do in arriving at their beliefs, it’s not propositional calculus; it’s not Bayesian probability; it’s not “believe whatever mama says”. But it is a little like all of those.

 ⤳⃝

I was riding on a train in Italy. Watching lemon trees out the window. Fantasising of tasting a lemon-based liqueur.

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Lemon trees. Amalfi Coast, Campania, Italy (color)

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My travel partner and I shared a vestibule with an American monk-cum-priest who introduced himself as Father John. Father John was making a pilgrimage from the Carolinas to Vatican City. I don’t know if he always evangelised but, although my partner and I tried to steer the conversation away from religion, Father John wanted to talk about his Catholic faith—specifically in a way that might score some converts.

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I don’t know whether the part of me that makes me debate with strangers online was acting out in its pre-internet form, or whether Fr J’s insistence on having a conversation we clearly did not want to have put me in a pugilistic mood.

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For whatever reason, I started querying him on some of the more outlandish assertions of Catholic doctrine. One thing I challenged him on in particular was transubstantiation.

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Try though the alchemists might they could never transmute lead to gold—but every Sunday around the world, holy men of Christianity transmute sacramental bread and wine into literally the body and blood of Christ.

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The biochemistry involved in going from wheat flour to bone marrow or from pectin to haemoglobin is not discussed in catechism, but the transition is obviously impossible by natural processes. Nonetheless, “the real presence of Christ in the Eucharist is a mystery—something so packed with meaning that we can never fully understand it.”

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I really don’t have a bone to pick with “transubstantiationalists”. I find the deeper reasons he and I think as we do more interesting than what we profess. I don’t go out of my way to attack people or hurt anyone’s feelings—but I do consider it rude to evangelise someone without consent.

So I needled the man. "Come on, you really believe that? Really? It’s not just a symbol? You can’t just have your religion without this physically impossible claim? Why would you insist on invoking the supernatural when that clearly undermines the credibility of everything else you say? Not only is it impossible according to science, even to your own sensory experience it just looks like a normal wafer—not like a hand or a butt or whatever. You literally, actually believe that this wafer literally, actually turns into actual human flesh of a dead man from two millennia ago—using up more body mass than he ever had all over the world every Sunday—really? Really?”

I still remember Father John’s response (which is how I’m able to tell you this story). He said: “OK, I understand your objections. But consider this. What if it were all true? What if the Resurrection, the Virgin Birth, G-d walking among men, the sacred mysteries, all of it were true? Wouldn’t that be wonderful? Wouldn’t that change everything about the way you see the world?”

What-if indeed, Father. What if.




Ever since I took too many mathematics classes, I started using the concept of “upper bound” literally. It confuses people.

  • Girlfriend: How long do you think it’ll take you to work out?
  • Me: Oh, I don’t know. I’d say less than five hours.
  • Girlfriend: Five hours?! What are you planning to do there?
  • Me: I didn’t say it would take five hours. I said it would take less than five hours.

Well, it did take me less than five hours. It took me an hour and a half.

 

Same problem with confidence intervals — I give these very literal answers. My former boss told me a story about a “rationality test” she was given, by a statistician or something. First she was asked to guess some fact that only a 5th grader would know, like how many tons the moon weighs or what’s the square mileage of Antarctica. Then the statistician asked her to give 90% confidence bounds. That’s, you’re 90% sure that the value is between these two numbers. Most people fail by saying numbers close to their original guess.

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My boss just said, "I’m confident that the number is somewhere between zero and a trillion trillion trillion trillion trillion trillion trillion … trillion." Well she was correct! And she was one of the only ones.

You can make confidence bounds as wide as you want and be logically correct. People will look at you in bewilderment when you say things like “I’ll be gone somewhere between two seconds and seven days,” but you will not be a liar.

 

Mathematicians are the only people who go around making statements like “I found out that the answer is greater than 6 and less than 3→3→64→2. I’m pretty sure the answer is 13, though.

(3→3→64→2 is bigger than billions of universes.)

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It’s not even the difference between certainty-of-proof and casual guesstimation. It’s the difference between giving an upper bound and giving the least upper bound (supremum).

I can say with complete confidence that I will never earn over 10^18737 pounds in my life, no matter how much hyperinflation or life extending medicines lie in the future. I can say the same about 10^18736 pounds. How low am I willing to go with these statements? Ay, there’s the lub.

 

No less of an intellect than Paul Graham swaps upper and lower bounds. In describing how he’s designing a new programming language (arc) with the goal that useful programs should be as short as possible in it, he writes:

That’s part of why I focus on code size. Length is an external constraint. If you start looking at code thinking "what is the lower bound on how long this has to be?" you’re one step from discovering the new operator that will make it that short.

I might be the only person who reads this and is confused. When I hear “lower bound” I think “Nothing is lower than the lower bound. It has to be bigger than the lower bound.” But then he is talking like putting a lower bound on the code means the code is shorter. Zoinks?

And then I’m like, oh. Duh. He means “the lower bound on the upper bound on how long this has to be.” Supremum. Well you could have just said that, Paul. Or I could not think so literally.