#math

[sukgiaquinto]

#dlg #fall winter #house plants #flight #maitland ward #flowerblr #starve #math #pullipdoll

[eat-heavy-point-drug]

#cute cats #fake friends #kpop smut #tokyo #gucci #swords #shoujo ai #math #newds for sale #desamor #lick my toes #sensory

[max1461]

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.

#math #navel gazing

[theconcealedweapon]

For many people, math is confusing in the theoretical sense but makes perfect sense if applied to a real world scenario.

For example, if you ask them what 25 times 13 is, they'll just see it as a big math problem. But if you ask what 25 cents times 13 is, they'll be like "that's easy, it's $3.25".

They may be confused by the new addition method being taught in school, where you add part of it to get a certain amount then add the rest. But if you ask them what 5 hours after 9:00 is, they'll be like "that's easy, 3 hours after 9:00 is 12:00 then you add the remaining 2 hours to get 2:00".

#math

[nonexistent-creature]

Math is fucked up because I will spend hours on a proof and then decide that it was easy. Like. No it wasn't! I was fighting for my life! Why?!?

#mathblr #math

[mathsuggestions]

Start a grad worker union

#submission #math #suggestion

[zacharylwackary]

if a bakers dozen is 13 then is a bakers gross 145, 156, or 169

#math

[msexcelfractal]

A project for a rainy day: automated long division in excel

#microsoft excel #my art #math #arithmetic

[mitchipedia]

#ephemera #math #computer science #funny #funny post

[sleepy-bebby]

#statistics #memes #meme #lol #mathematics #mathblr #math #maths #relatable #relatable meme #relatable memes #dog meme #dog memes #this is fine meme #this is fine dog #funny #Twitter #Twitter meme #Twitter memes

[demonicseries]

You guys are never gonna believe what the name of this sculpture is

#math #Loki #lokius #Mobius #Mobius m Mobius #mathematics #topology

[mostly-funnytwittertweets]

#history #math #twitter #meme #memes #tweets #tweet #funny #lol #humor

[max1461]

All I do lately is read nLab and watch mario maker streams and it's fucking with my brain. Can mario be viewed as a category?

#math #mario

[prokopetz]

People on this site will put together polls like "The Banach-Tarski Paradox versus Camembert Cheese", then act like the results prove that they're surrounded by idiots.

#tumblr #polls #mathematics #math #banach-tarski paradox #food #cheese

[knotty-et-al]

Visualization of the Rubik's cube

#permutation #permutations #groups #group theory #rubiks cube #rubikscube #math #mathematics #math visualization #visualization #knottys math #math stuff #mathy stuffy #cube #hexahedron #combinatorics

[iamnotaware]

ok wait, reblog if you’ve cried at least once because of math, doesn’t matter which grade i’m trying to prove something 

#math