Posts tagged with logic

Both direct sum and tensor product are standard ways of putting together little Hilbert spaces to form big ones. They are used for different purposes. Suppose we have two physical systems…. Roughly speaking, if … a physical system’s … states are either of A OR of B, its Hilbert space will be [a] direct sum…. If we have a system whose states are states of A AND states of B, its Hilbert space will be [a] tensor product….
MEASURE SPACE   disjoint union  Cartesian product
HILBERT SPACE   direct sum      tensor product

John Baez

(Source: math.ucr.edu)




Scepticism trades on [1] a focus on the worst case, and [2] a demand that any method of forming belief find the truth in all logically possible circumstances.

When action must be taken, scepticism is in league with obscurantism, with know-nothingism, and in opposition to forces that are more optimistic about the information that inquiry can provide to judgement.
Clark Grymour

(Source: hss.cmu.edu)




This is a question about argument, counterargument, convincing people of something, and why people believe the things they do.

  • Let’s say you make a claim. For example the claim that rich people are rich because they do the most good in society.
  • I want to argue that you’re wrong. There are a couple ways I could proceed.
  • Now, to me, personally, the most logical tack to take should be to ask you for evidence.
    Wikipedian Protester

    You’ve made a sweeping claim about a large number of people, using ill-defined abstractions like "good", and so on.
  • In my mind, the way I personally think, I should ask you to back that up, you won’t be able to (or will make recourse to doctors, neglecting the real issues like LBO investors or the bottom billion) and then you end as wrong or neutral.
  • But. This is not what really works in a real debate or argument. It’s mysterious to me as to why, but I think a correct answer as to why would be pure gold.
  • What’s going to work better is if I argue a separate theory.  Like "No, the rich don’t benefit society the most, they just screw people over the most. They put the terms of trade in their favour, they set up deals that screw over powerless or uninformed parties, and the game is set up in a way that benefits them.” If I’m really impassioned and present some stories backing up my viewpoint this will work even better.
  • Now to me this seems illogical. You’re saying "X" and rather than responding "Not X" I’m supposed to respond "Y". Y relates somewhat to X and kind-of negates X, but mostly Y is just a different theory of the world.
  • Another example could be that you argue for supply-side economics. Instead of me arguing against supply-side economics, pointing out the flaws or weak points in it, it’s more convincing if I argue Keynesianism, or MMT, or some other full theory, instead.

My examples are economics debates because I’m stupid. But this principle of arguing in a different direction than directly contrary to what you say works elsewhere too.

  • Look up "Gish Gallop" for example. The phrase relates to evolutionists complaining about the way a creationist, Duane Gish, argues. Gish allegedly adds more and more propositions to his argument, forcing his opponent to look up and refute stuff much slower than Gish can add new propositions.
  • Gish is not arguing this way, and his evolutionist opponents are not frustrated by the rhetorical style, because it doesn’t work. Irrespective of how much he does it, the fact that evolutionists were bothered enough to name a “fallacy" after Duane Gish indicates that audiences were swayed by the technique of adding more and more propositions.
  • Logically—to me at least—it’s harder to prove a claim made up of many propositions than to prove just one of the propositions making up the claim. So “Not only does G-d exist, but the Christian G-d exists, and was made manifest as Jesus Christ, and died on the cross to atone for the sins of mankind, and ten other points of doctrine" —- should be harder to prove than just "Any G-d exists”. But yet I’ve seen more than one “Atheism debate” where the anti-atheist person debates this very long proposition.
  • Or let’s say you’re arguing that a thesis you read in the Times is “probably right" because it’s vetted by experts. I should argue back that vetting doesn’t imply it’s right. But to be more convincing, I probably should counter the Times writer’s theory with one of my own. If I don’t have one, but just like to carry around a bucket of scepticism to pour on fires of passion? I’m SOL rhetorically.
  • Why wouldn’t you just defend the easiest argument—the one that Pareto-dominates the long argument?
  • The fact that people don’t agree with what I’m calling simple logic means I’m missing something. In fact I don’t think anyone has a theory of why people are convinced by things, which captures the appeal of these run-on arguments. Of course I would be happy to be told I’m wrong about that; please tell me if I am.
  • Do people prefer more information-dense statements? Does making a more specific claim imply, in some wider "ecological" sense, that the speaker is “more likely to be” well-informed? Do people prefer whole frameworks to piecemeal facts? If so, why?
  • I could go on with more questions and half-baked theories of what might be happening, but I’ll spare you.

So, why do people think this way? Is it a lack of sfumato? And what does the fact that people think this way tell us about other important stuff, like rationality, love relationships, parenting, reasoning, good decisionmaking, "facts", habits, authority, marketing, judgement, court convictions, investing/retirement planning, political voting, how people come to their beliefs, and what it takes to change someone’s beliefs?




supplysideliberal:

[M]y only qualifications … are having read several books about marriage and experience from my own marriage, now in its 30th year…
  • “… a misconception: … to make a marriage work, you have to find the right person. The fact is, you have to be the right person,…
  • … students … interview friends about their … weaknesses, and discuss what triggers their … reactions … in order to understand their … hot buttons… “Being blind to these causes people to experience problems as due to someone else—not to themselves,” Solomon explains. “We all have triggers, blind spots, growing edges, vulnerabilities. The best thing we can do is be aware of them, take responsibility for them, and learn how to work with them effectively.”

  • … blaming, oversimplifying, and seeing themselves as victims are all common traits of unhappy couples and failed marriages. …
  • frame statements as “X, Y, Z” statements, rather than finger pointing: When you did X, in situation Y, I felt Z….

Here are a few of my own thoughts on marriage: 

  1. There are a huge number of dimensions on which one might wish to be well-matched with one’s spouse. There is no way you are going to be well-matched on all of those dimensions.
  2. The reputation you have built up with your partner for telling the truth about objective facts is a precious asset in any relationship. …[T]here is bound to be some way to tell the truth. (If you can’t think straight, say “I can’t talk about this right now,” rather than lying.) The more subjective realm of revealing what is in your heart is trickier; seize moments when you will be able to express yourself well and be well understood. It is worth working toward being known.
  3. In an argument, if each partner comes back with 101% of the irate heat the other just gave, things will explode. But if each partner ratchets down the intensity to 99% of the intensity of the last remark, things will eventually calm down.* So a small difference in reaction pattern can be the difference between an explosion and something that simmers down.

    

* Math note: To pursue the logic a bit more, if your partner is coming back with 125% intensity on each round, you are going to have to return less than 80% intensity on each round to avoid an explosive chain reaction….

A regular reader writes in:

I tried that for a long time. It was better than escalating in the short term. But I eventually realized that some people like explosions, and created them when they weren’t happening.

…having been married for a long time, I can guarantee you, that no one on the outside has any idea what goes on inside a marriage.

The parts of being married I enjoyed were great. I’ll never do it again.




The tropical semiring is arithmetic piped through a log with base →.

Also if you or someone you know  is first encountering a squeeze theorem or other a≤x≤A type reasoning, remark 2.1 might be a relatively painless calisthenic to warm you/them up to a≤x≤A type arguments.

by Gregory Mikhalkin










1. The monad, of which we will speak here, is nothing else than a simple substance, which goes to make up compounds; by simple, we mean without parts.

2. There must be simple substances because there are compound substances; for the compound is nothing else than a collection or aggregatum of simple substances.

Gottfried W. Leibniz, Monadology

 

Crazy how a “father” of calculus was so illogical in his seminal work of 1714.

  • The existence of compound things does not imply the existence of partless atoms.
  • He asserts, doesn’t prove, that a compound is “nothing more than" a collection of simple substances. (atoms)







I’ve collected a few tidbits about non-wellfoundedness on isomorphismes:

  • the opposite of the idea of “indivisible atoms" at the "bottom" of everything
  • turtles all the way down
  • (infinite regress is OK)
  • a > b > c > a
  • (so the two options I can think of for non-wellfounded sets are either an infinite straight line or a circle—which biject by stereographic projection)

as well as examples of irreducible things:

  • if you take away one Borromean ring
    http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Borromean_Rings_Illusion.png/774px-Borromean_Rings_Illusion.png
    then the whole is no longer interlinked
  • Twisted products in K-theory are different to straight products.
    https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/M%C3%B6biusStripAsSquare.svg/1000px-M%C3%B6biusStripAsSquare.svg.png
    A Möbius band is different to a wedding band.
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    Yet 100% of the difference is in how two 1-D lines are put together. The parts in the recipe are the same, it’s the way they’re combined (twisted or straight product) that makes the difference.

So Leibnitz’s assertions are not only unsupported, but wrong. (Markov, causality, St Anselm’s argument, conservation of mass, etc. in Monadology 4, 5, 22, 44, 45.)

tl,dr: Leibniz, like Spinoza, uses the word “therefore” to mean “and here’s another thing I’m assuming”.




Walter Ong turns to the fieldwork of the Russian psychologist Aleksandr Romanovich Luria among illiterate peoples [of] Uzbekistan and Kyrgyzstan … in the 1930’s.

Luria found striking differences between illiterate and even slightly literate subjects, not in what they knew, but in how they thought.

Logic implicates symbolism directly: things are members of classes; they possess qualities, which are abstracted and generalised.

Les Mots & Les Images by René MagritteOral people lacked the categories that become second nature even to illiterate individuals [living] in literate cultures…. They would not accept logical syllogisms.

A typical question:

—In the Far North, where there is snow, all the bears are white.

—Novaya Zembla is in the Far North and there is always snow there.

—What colour are the bears?

—I don’t know. I’ve seen a black bear. I’ve never seen any others…. Each locality has its own animals.

….

"Try to explain to me what a tree is," Luria says, and a peasant replies: "Why should I? Everyone knows what a tree is, they don’t need me telling them."

James Gleick, The Information, citing Walter J. Ong and Aleksandr Romanovich Luria

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from http://en.wikipedia.org/wiki/William_Blake#Royal_Academy:

Over time, Blake came to detest Joshua Reynolds attitude towards art, especially his pursuit of “general truth” and “general beauty”. Reynolds wrote in his Discourses that the “disposition to abstractions, to generalising and classification, is the great glory of the human mind”; Blake responded, in marginalia to his personal copy, that "To Generalize is to be an Idiot; To Particularize is the Alone Distinction of Merit".[20]

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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”.)

File:Catechism-Madras Presidency Village.jpg

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.

File:Jules-Alexis Muenier - La Leçon de catéchisme.jpg

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.

File:Pietro Longhi 021.jpg

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.




horizontal composition of homotopies
within 𝒵:
red→yellow then orange→green
ends the same as
orange→green then red→yellow

horizontal composition of homotopies

  • within 𝒵:
  • red→yellow then orange→green
  • ends the same as
  • orange→green then red→yellow

(Source: youtube.com)


hi-res




100 Plays • Download

Discussing the first year of life.

  • In-born aptitudes include
    • theory of other peoples’ minds (looking at where they’re looking, show surprise when they know something they “shouldn’t”) (this gives some idea of what autistic people go through—they lack a skill that one-year-olds have!)
    • animate versus inanimate things—causality, essentially

Infants do reason. They just have less knowledge than adults.

Brain hemispheres act the same at birth but begin to specialise over time. (So could there be an alien environment where human babies would specialise their brains along different lines?)

There’s no one thing that makes the human brain superior to other animals’ brains (at least not that we’ve found). It’s thought to be the interaction of various factors—as well as our long developmental period—that make us able to build rocket ships, paint potato eaters, invent radio and discuss these things on it.

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(Source: BBC)