m---a---x - Tumblr blog

m---a---x · 4 hours

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Just did, ended up back home. Luckily i can't fly to space

Take a walk, randomly.

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m---a---x · 5 hours

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This is what it's like proving an interim lemma

I'm not too proud to admit that I desperately crave the safety and security of being a distributed hive mind which cannot be feasibly stamped out by any amount of state exerted power. I'm also not too proud to ask for help. Please join my collective, please merge into me and become one of the manifold drones which jointly make up my greatness.

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m---a---x · 9 hours

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Well, usually it means that the definition produces an object of the required type.

So in case of a function one needs to check that it is left complete and right unique. Or maybe it is supposed to be a continuous function to a compact subspace, then one needs to additionally check whether it is continuous and it's image precompact.

Or maybe the defined object is not supposed to be a function at all, but a tree. Then one needs to check all the tree properties, that being (depending on what notion of tree one is working with) cycle-free and connected.

kinda hate the questions that are like prove this mathematical object is well defined i never know what im actually meant to prove there (i guess it’s like, for example function returns a value in the range, doesn’t give two different value for equivalent input, etc??)

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m---a---x · 11 hours

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I wish there were non-religious monastaries. A math monastery would be so cool, I'd immediatly join.

How much do you think one would need in investments to live off passive income where one buys some thousand dollar plot of land in New Mexico and lives in a storage shed? Just enough for food and medicine right? There are cheap plots of land in walking distance of places you can buy food and medicine. I think about this pretty much every day. Ever since I started investing in an index fund around a year ago. Would the suck of not having electricity and plumbing (well presumably I would be able to charge up my phone and laptop on grocery runs) be more or less than the suck of having to work for a living? I have enough invested for my fifty dollar a month medicine, just not for food yet...

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m---a---x · 2 days

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Ah i see! Very interesting. I never took any homotopy theory classes, this sort of stuff makes me whish i did!

There he is. There’s my boy!!!

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m---a---x · 3 days

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Yeah, but I think that most people want to criticize unwillingness to consider other views or arguments whe they say

"You always act like your opinions are correct and everyone else is wrong"

As in, beyond doubt or in generality above others

Something I always find bizarre is when people mention something along the lines of "You always act like your opinions are correct and everyone else is wrong" as like. A moral or personal flaw. Because like. I'm pretty sure that's just how opinions work.

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m---a---x · 3 days

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Something I always find bizarre is when people mention something along the lines of "You always act like your opinions are correct and everyone else is wrong" as like. A moral or personal flaw. Because like. I'm pretty sure that's just how opinions work.

#yeah #this as well

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m---a---x · 3 days

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When looking at pthers we see that basically everyone has some wrong opinions. So statistically that should hold for ourselfes as well. Thus, while we consider each of our opinions as correct, this means that we are wrong on some of them. But, since we do not know which, we must consider every one of our convictions with some measure of doubt. Of course not of equal measure for all of them and we even consider ourselfes as one of the most correct people, but nontheless, doubt should remain. Hence we need to listen to the arguments of others, reevaluate our conclusions and be ready to admit when we are wrong.

When someone does not do this, it is fair criticism. While we may consider our opinions as correct we should never do so to the absolute.

Something I always find bizarre is when people mention something along the lines of "You always act like your opinions are correct and everyone else is wrong" as like. A moral or personal flaw. Because like. I'm pretty sure that's just how opinions work.

#Only a sith deals in absolutes

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m---a---x · 3 days

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getting surgery to give me a second head so i can tell twice as many lies as usual

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m---a---x · 4 days

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Maybe what we need to fix the "being forced to publish at a ridiculous pace" problem in irl academia is to get more quasi-immortal beings to whom time isn't an object among university higher-ups

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m---a---x · 5 days

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Allways all this speeping under the carpet. Using second order doesn't give more knowledge, it just definitionally moves the unknown from truth to "properties"

I believe that the following philosophical argument in favor of the second order Peano axioms as ultimately "correct" works:

We know from Gödel that no effectively definable formal system can capture the full behavior of the "true" natural numbers. That is, it's impossible, as finitistic beings, to give a formal definition which precisely characterizes the standard natural numbers. We will always "leave out some details" in the definition, among these the Gödel sentence in the given system and so on.

This makes the meaning of the phrase "the standard natural numbers" itself philosophically problematic. In the context of a given meta-theory (say ZFC), we can take the standard naturals to be some particular meta-theoretic construction (say, the von Neumann ordinals). In this context, the incompleteness theorems as internalized in the meta-theory say that no effectively definable formal system as internalized in the meta-theory can prove all the true facts about our chosen standard model. But of course this doesn't save us, because the incompleteness theorems "on the outside" of the meta-theory say that it can't prove everything there is to know about the "true" external standard model of the naturals, whatever it is.

Of course this last part is possibly bullshit and may rely on some kind of Platonism to make sense. So to be a conservative as possible one should stick to just asserting the meta-theory-internal version of the incompleteness theorems. After that you can, if you want, let them inspire by implication a sort of fog of uncertainty in the reader about what fucked up epistemic shit is going on "outside" the meta-theory, even though that perhaps does not make sense (or perhaps it does...). Of course you can make "outside the meta-theory" make sense by internalizing the meta-theory in a meta-meta-theory, but then you just get the same situation one level up.

So, ok, the point is that you are never going to be able to write down a formal system that unambiguously defines what you mean by "the true standard model of the naturals", such that exactly the statements which can be derived from this system (=definition) are exactly the true ones. Which sucks! That's lame, because math is supposed to involved being precise about what we mean by shit.

There are a couple of ways out. One is to just take some effectively definable formal system like first order PA and say "this is what we mean by the naturals, we mean the shit that can be proved from this. Yeah that leaves a lot of stuff hanging, a lot of statements about arithmetic of-ambiguous-truth-value, but whatever". Because, you know, PA is not categorical, so it has many inequivalent models. Or you can say "I will take second order PA as internalized in ZFC (so basically, the von Neumann ordinals) as my definition of the naturals". Which I think is more powerful(?) but still suffers from the same problem when you look at it "from the outside" of ZFC. Actually, you can do that for any (expressive enough) meta-theory M, you can put second-order PA inside it and take that as your naturals.

With the stage set, a brief digression:

I think that, informally, we should all be able to agree on the following about the "true" set of natural numbers, if such a thing can be said to exist (and imo it sort of must, because it's implicitly invoked in a meta-way when we define formal systems to begin with, and so on):

1. The number 0 is a natural number 2. If n is a natural number, then the successor of n (that is, n+1) is also a natural number 3. If m and n are two natural numbers and they have the same successor (that is, n+1 = m+1), then m = n 4. There is no natural number whose successor is 0 5. If P is some property which might or might not hold of a natural number, and we know that P holds of 0, and we furthermore know that whenever P holds of one number it must hold for the next number, then we know that P must hold for every natural number

Some people are philosophical uncomfortable with the last one, but I think it's intuitively undeniable. Like imagine a fucking... guy hopping from one number to the next, and he never stops. Can you pick a number he never gets to? No you fucking can't. You believe in induction.

So, ok, back to models and shit: both first order and second order PA try to formalize this intuition, and the key way that they differ is in terms of what a "property" (mentioned in (5)) is. First order PA says that a "property" is a first order formula. This is very powerful because we can effectively define the set of first order formulas over a given language. They are finite objects and we can work with them direction. From this flows all the nice properties of first order logic, like completeness and so on. But this effectively definability also makes it susceptible to the incompleteness theorems, and so first order PA ends up "leaving stuff out".

Second order PA defers the notion of a "property" to the meta-theory. It basically says "a property is whatever you think it is, big guy ;)" to ZFC or whatever theory it's being formulated in. ZFC thinks a property is a ZFC-set. Meta theory M thinks a property is an M-set. And second order PA as formalized in M agrees. Mathematically this makes second order PA harder to study as an object in itself. But philosophically I think it's kind of desirable?

First of all because, at a basic level, "property" seems like a much more fundamental notion to me than "natural number", and one I am much more willing to accept an intuition based definition of. Like, I don't know what you mean if you say "the true natural numbers". That seems pretty wishy-washy! But if you say "the real-world, ordinary definition of 'a property'", I can kinda be like "yeah, properties of things. I know how to reason about those!". And then second order PA, because it's categorical, will tell me "great: since you know what a property is, here's what a natural number is". And that's something I can work with.

This was overly long-winded I think. But in other words, what I am basically advocating for is conceptualizing second order PA as a function from "notions of property" to "notions of the natural numbers". And because models of PA are unique up to isomorphism (in whatever (sufficiently powerful) meta-theory you formalize it in, not "from the outside" of course) this means you can take up SOPA as your definition of the natural numbers and then "lug it around with you" into whatever different foundational system or meta-theory you fancy. And when you lug it into the real world, where "properties" mean actual properties of things, you get the real, true natural numbers.

This is all purely philosophizing of course. But I think this is about the situation.

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m---a---x · 5 days

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Buy a spaceship to absolve you of your sins

an AAAAA game can't just be "game we put a lot of money into", that's not enough to kick it into the next bracket. the development needs to be on the level of medieval cathedrals: spanning multiple generations and involving hundreds of accidental deaths

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m---a---x · 5 days

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Star Citizen

an AAAAA game can't just be "game we put a lot of money into", that's not enough to kick it into the next bracket. the development needs to be on the level of medieval cathedrals: spanning multiple generations and involving hundreds of accidental deaths

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m---a---x · 5 days

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rpg setting with multiple competing units of damage/resilience used in different regions. you gotta worry about the conversion between hp celsius and hp fahrenheit

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m---a---x · 7 days

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download killing upload pain. instant thousand deaths to brain. motherboard on murder spree. blood computer victory.

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m---a---x · 7 days

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No, i fucking love Haskell :P (Though there are ofcourse many limitations, for example slowness which you can't really fix if you don't write extensive ways to compile fixedpoint recursion as iterations)

What would you like to see in a new programming language? I'm feeling in the mood to write a compiler again, but I am not sure what sort of language to make this time.

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m---a---x · 7 days

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Oh, my whishlist would be true functional with a good high level library which doesn't try to be imperative (looking at you, lisp)

What would you like to see in a new programming language? I'm feeling in the mood to write a compiler again, but I am not sure what sort of language to make this time.

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