Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Intuitionism

en.wikipedia.org

Post reply on HN
Print view View original

81–90 of 179 posts

Re: Intuitionism

#81

Post by circlefavshape »

RubyRidgeRandy wrote: post #65
circlefavshape wrote: post #18

Earlier quoted context omitted.

I've been thinking about this recently, and realised that your framing here casts humans as separate from the rest of reality. Your sense organs and your brain are part of things-in-and-and-of-themselves.

I don't think it does. Humans are agents within reality and have perceptions of reality. Your brain having a representation in this reality that might be different from 'true reality' doesn't change the argument at all.

I don't see how your perceptions can be anything other than a direct experience of reality interacting with itself unless you imagine that your mind is separated from reality somehow

Re: Intuitionism

#82

Post by roywiggins »

Tainnor wrote: post #56
constantcrying wrote: post #31

Earlier quoted context omitted.

>I think most here would know that math is not complete, consistent or decidable. There is zero evidence ZFC is inconsistent. Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones. Not even hardcore Platonists would claim that the number 1 exists in physical reality. But…

There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?). ZFC is weird (especially choi…

I would bet that, even if ZFC is not consistent, there's another set of axioms which is, and in which all the stuff we've proven in ZFC still holds. That is, ZFC just happens to be a useful framework for the mathematics we're interested in. Even if ZFC collapses it seems very unlikely that all the stuff we've proven within it will; instead, we'll fix ZFC, like ZFC "fixed" naive set theory.

Re: Intuitionism

#83

Post by rambojohnson »

kshahkshah wrote: post #71

The older I get the more I feel like pursuing these trains of thought will lead to mental instability and sadness. We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true. Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here

I understand where you're coming from, and pragmatic perspectives certainly have their place. However, philosophy, including intuitionism, can offer us new ways to comprehend the world. It's not about destabilizing our understanding, but rather enriching it.

Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.

Re: Intuitionism

#84

Post by athrowaway3z »

trabant00 wrote: post #25

I think most here would know that math is not complete, consistent or decidable. ( https://www.youtube.com/watch?v=HeQX2HjkcNo ) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life. My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's im…

My looking at natural numbers gave me the exact opposite conclusion.

Once the grey blob of infinite nothingness becomes distinct enough that there is a difference between one moment/point and another moment/point, you have enough to start counting different states (or do binary arithmetic). Its high school math to find out some numbers are different then others and eventually you'll find the primes.

I'll bet there is no god, power, or alternative rules in any possible universe, fictional or real, that could not find the primes.

Re: Intuitionism

#85

Post by smokel »

pixl97 wrote: post #67
smokel wrote: post #37

Earlier quoted context omitted.

Thank you for your reply. I'll try some more! :) I noticed that in nature (i.e. in the physical universe) there appears to be no logic. There are not even two things exactly the same, as far as I can tell. So that lead me to believe that "counting" (or abstraction) is not even a property of the universe, but possibly only a human (or animal, or Turing machine) construct. To me, logic only starts to occur when a very…

>There are not even two things exactly the same, as far as I can tell. https://medium.com/physics-as-a-foreign-language/how-do-we-k... https://www.popularmechanics.com/science/news/a27731/what-if...

Your url-based reply is quite ironic.

The first link is an article that explains that all electrons are exactly identical, suggesting that there are indeed things in the universe that are exactly the same. However, the second link discusses the one-electron idea by Wheeler, which suggests the exact opposite :)

Re: Intuitionism

#86

Post by sunsetdive »

This is post-modernism applied to math. It concludes in solipsism.

Since the mind and its mental constructs are a part of the objective reality, they will end up describing aspects of objective reality. If they don't, they break down, become chaotic and incomprehensible to those grounded in the objective reality.

Re: Intuitionism

#87

Post by glcheetham »

rambojohnson wrote: post #83
kshahkshah wrote: post #71

The older I get the more I feel like pursuing these trains of thought will lead to mental instability and sadness. We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true. Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here

I understand where you're coming from, and pragmatic perspectives certainly have their place. However, philosophy, including intuitionism, can offer us new ways to comprehend the world. It's not about destabilizing our understanding, but rather enriching it. Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we…

Sounds like GPT wrote this comment

Re: Intuitionism

#88

Post by derefr »

ttctciyf wrote: post #20
smokel wrote: post #2

I have liked intuitionism from the very moment I first heard about it. I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences". It may also help to guide us away from the "why is there something rather than nothing" problem. If existence is total chaos, then we a…

> fools us into thinking there is some inherent order -- which there isn't. Bold claim :) Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other? But if mathematicians have this ability,…

In other words, the Cartesian principle extended to two parties?

Cogito sicut tu cogitas, ergo sum sicut structus es: I think as you think (at least for a moment); therefore, I am, at some level, structured as you are (at least for a moment); and therefore, there is order in the universe, at least in the temporary alignment of our structures that enables us to think along these same lines?

(Saying nothing, of course, of whether you "exist." I may be a brain in a vat, and your thoughts may be the emission of an evil demon running a simulation — but that still means that my thoughts and the evil demon's simulation share structure that implies an underlying shared set of computational axioms!)

Re: Intuitionism

#89

Post by ttctciyf »

klik99 wrote: post #68
ttctciyf wrote: post #60

Earlier quoted context omitted.

> built on top of chaos. I don't think you've taken on the full force of the argument that regularity in human activity (such as building systems ) requires a source of order for it not to simply dissolve into chaos itself. > How can it be that the patterns that we observe are so consistent? How can we claim to discern consistency (or inconsistency) without the ability to follow a rule correctly? And how can we follo…

This argument completely ignores the observer which is bound by the same limits of our processing - indeed they are paired together. Entering purely theoretical space here: On the timescale of eternity this might be a local pocket of some logical organization but there is no fundamental logic governing everything. Our observation is limited so we can’t perceive chaos, instead evolved to only recognize patterns. Over…

Well, not a possibility I considered, but nothing in the argument depends on the regularity being a permanent feature of the cosmos, just that systematic human endeavours, such as mathematics or indeed meaningful debate, depend on it, so when it goes they go. In that sense, if you wish to consider this conversation meaningful you are kind of ceding the point that for now chaos doesn't, in fact, reign. If you don't consider it meaningful, then why are you having it? :)

Re: Intuitionism

#90

Post by BSEdlMMldESB »

practal wrote: post #48
BSEdlMMldESB wrote: post #44

Earlier quoted context omitted.

if the system is sound, then (I think) by definition you cannot prove different (wrong) conclusions. if you derive a different result, by soundness those would be equivalent ???

It does not really matter if the system is sound or not, right? Although of course a sound one is far more interesting. Anyway, any way of justifying this is mathematical (and so would be the definition of soundness, if it was relevant here). If math is not real, then there is no justification.

math describes (fragments) of reality;

therefore it is of no consequence if math as itself is "real" or not. it is intended to model whatever "real" even is.

somewhat similarly: in modern logical theories whatever "true" (and/including "false") even mean doesn't matter. is left out of the logical theory and it is effectively a mere parameter.

all the subject does is gurantee "truth in, truth out" (and complementarily "false in, false out")

the precise details of true "and/including" false, seems to me, are somewhere in the boundary between "classical" and "intuitionism" (or "constructivism")

the subtle distinction between intuitionism" and "constructivism" is above my pay grade (and seemingly above the paygrade of everybody I've had the chance of discussing this with)

Post reply on HN