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Introduction to Introduction to Category Theory

My masters degree involved a good bit of category theory. Personally, I don't see how it has any use outside of mathematics. (Note 'maths' includes 'mathematical logic' - so it's still a broad field of applicability).

I am highly motivated to be persuaded otherwise, and hence will be watching this series of posts with keen interest.

---
<disclaimer>
I am not a working mathematician, and have not published any papers. My masters thesis involved a lot of category theory - but only relatively simple category-theoretic concepts (it was an application of category theory to a subfield of mathematical logic).

Limits, free objects, adjunctions, natural transformations etc. but not higher-order categories, topoi/toposes or anything fancy like that.
</disclaimer>

<handwavy discussion of technical math>
As I understand it, the usual application of category theory is mostly to things involving natural transformations (it is said the need for a way to formalize natural transformations is what led to the invention of category theory) - and even then, it seems mostly to be applied to nice algebraic objects with (category theoretic) limits, and slight generalizations of these. So, to groups and rings and modules, and then to some categories made of stuff kinda like those things.

There's also the connection to topology and logic, via simplicial complexes, homotopies, toposes, type theory etc. which seems very interesting to me. It seems useful if you want to think about constructive mathematics (i.e., no law of excluded middle) - which is promising for maths involving some notion of 'computation' (for a given abstraction of computation) which has obvious applications to computing (especially automated theorem provers / checkers).

In these senses, I can certainly see its use as another mathematical field, and a good way of reasoning IN MATHS or ABOUT MATHS. But I don't quite understand its tremendous reputation as this amazing mathematical device, and less so its applications outside of what I've mentioned above.
</handwavy discussion of technical math>
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In particular, from my admittedly limited knowledge, category theory only seems useful:
a) If you already have a bunch of different fields and you want to find the connections.
b) If you want to start up a new field, and you need a grounding (after which the useful stuff will be specifically in the field being developed, and not a general category theory result).
c) For good notation / diagrams / concepts for a few things.

EDIT: Interested to hear the opinion of someone who actually works with category theory on a regular basis.

Is Scott Alexander bad at math?

I find that what helps for me is re-writing maths as I'm learning it.

When I glance at an equation or formula (especially an unfamiliar one), I usually can't take it in because my mind is trying to glance it all at once. I have to force myself to scan it slowly, either by re-writing it, writing out its definition, or by (holding a ruler under it) and scanning one symbol at a time.

Then again, I'm currently studying a postgraduate degree in maths and I'm not someone who's ever considered themselves 'bad at math'.

Rationality Quotes Thread May 2015

This comment resonates with me. I am also a Christian-turned-Atheist.

When something bad happens, or I feel in danger, or I don't know what to do, usually I want to send up a prayer. Then I have to catch myself and remember that yeah, that's not going to help.

The path of the rationalist

last paragraph teaches false lesson on cleverness

What exactly do you believe the false lesson to be and why do you think it's false?

I interpreted it as meaning one should take into account your prior for whether someone with a gambling machine is telling the truth about how the machine works.

What's in a name? That which we call a rationalist…

So much for "the map is not the territory", I guess.

Defecting by Accident - A Flaw Common to Analytical People

That's a useful template and in some cases the advice...

This may vary somewhat with the audience and I believe the claim...

Note, that I did notice the change. I do think that to facilitate proper understanding of a sentence, 'but' should be used slightly differently from 'and', even if both are technically correct.

Defecting by Accident - A Flaw Common to Analytical People

So, Viliam_Bur, do I understand correctly?

You are saying the major tradeoff isn't between:

  • Speak 'bluntly' in situation X
  • Speak 'politely' in situation X

It is between:

  • Speak 'bluntly' in every situation (default)
  • Invest effort to learn to speak more 'politely'

(The costs-benefit calculation is a long-term one performed over all potential situations, not a short-term one performed over each specific situation)

I agree; this makes sense to me.


In certain cases, bluntness can be useful. However, by this I mean it can be useful if you are able to let people be blunt to you. See Crocker's Rules and the related article on Radical Honesty.

If everyone in a certain social context operate on such a system (whether explicitly or implicitly), then there is some benefit to these people in terms of saving time and cognitive effort in the short term, and in the long term if they haven't yet spent time on developing 'politeness'.

Defecting by Accident - A Flaw Common to Analytical People

I also recall reading 'and', if not in that book then in one on a similar topic.

I believe the basic format for using 'and' is: "I believe X is good, and it could be even better if you did Y".

Contrast:

  • "Your speech was good, but consider using more specific examples"
  • "Your speech was good. However, it could be improved with more specific examples."
  • "Your speech was good. Yet I think that using more specific examples would improve it."
  • "Your speech was good, and I think you could increase the impact even further if you also included more specific examples."

(Note: The one with 'yet' sounds a bit awkward to me, I'm not sure I know how to use it in this situation).

Sure the use of the word 'and' is neither neccessary nor sufficient to make the sentence more positive, but I think that (given a bit of practice) it naturally causes you to do so. Much the same as the word 'yet', but (I think) more strongly.


I could theoretically say "Your speech was good, but I think you could increase the impact even further if you also included more specific examples.", but using the word 'but' doesn't really force me to do so the way that using 'and' would, and doesn't come across as quite as supportive. The word 'but' actually sounds slightly wrong to me in this sentence.

Rewriting the sequences?

I'm not sure Twelve Virtues of Rationality is the best place to start. To be honest, I was a bit confused reading it the first time, and it only made sense to me after I had spent some time on lesswrong getting used to Eliezer's writing-style.

For myself (as I know it was for many others), I got here via Harry Potter and the Methods of Rationality. I'd say it's a great place to start many people off, but perhaps not the majority. Along with that, what got me convinced to start reading lesswrong was my interest in biases and importantly being convinced that I, myself, am biased.

Thus I would propose one starts off with a single post about some bias, especially one that convinces the reader that this is not an abstract experiment involving random test-subjects. I think that Hindsight Devalues Science works excellently for this purpose, although it's obviously not written as an introductory essay.

Follow this up with some posts from Map And Territory, namely: What Do We Mean by Rationality, What is Evidence, and The Lens that Sees its Flaws, in that order, to give a basic introduction to what rationality actually is. This could be followed by one or two more posts from Mysterious Answers to Mysterious Questions, so why not start with the first two: Making Beliefs Pay Rent in Anticipated Experiences and Belief in Belief.

Now, you could finally digress to Twelve Virtues Of Rationality and then maybe try your hand at the whole Map and Territory Sequence (skipping over those posts you've already seen), alternatively you could finish reading the Mysterious Answers to Mysterious Questions sequence first.

After this, I no longer provide any advice as to reading order. You could choose to follow the order provided by XiXiDu above. I provide the following as one order which would at least do better than picking articles at random:

Finish reading Mysterious Answers to Mysterious Questions if you haven't already done so.

The whole mega-sequence of How to Actually Change Your Mind contains a lot of pretty important stuff, but will take a while to read.

The rest of Lesswrong. ;)


Summary:

And then follow with either of:

Path a

Path b


Which concludes my recommendation.

Post ridiculous munchkin ideas!

Relevant to this topic: Keith Johnstone's 'Masks'. It would be better to read the relevant section in his book "Impro" for the whole story (I got it at my university library) but this collection of quotes followed by this video should give enough of an introduction.

The idea is that while the people wear these masks, they are able to become a character with a personality different from the actor's original. The actor doesn't feel as if they are controlling the character. That being said, it doesn't happen immediately: It can take a few sessions for the actor to get the feel for the thing. The other thing is that the Masks usually have to learn to talk (albeit at an advanced pace) eventually taking on the vocabulary of their host. It's very interesting reading, to say the least.

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