Intuitionism • Hacker News Board

Re: Intuitionism

#111

Post by zoogeny » Fri, Jul 14, 2023, 5:37 PM UTC

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,…

> mustn't this in turn depend on some pre-existing regularity

Consider the set of all things, including the ordered and disordered. Consider the operation of taking subsets of that set. Consider that conscious entities such as us only arise in ordered subsets (for some definition of ordered).

Those conscious entities would see their proximal environment as ordered. They might assume that the only things in the set of all things are those things that are ordered in the same way as the proximal environment from which they arose.

Our universe may be just a subset of a subset. That is, our universe might be one kind of universe that has some apparent quantum-field or relativity-geometry based regularity. There might be some number of universes that share that kind of ordering and perhaps not in all of them conscious entities arise.

It is still an interesting question to ask: why would conscious entities arise within this particular kind of ordered universe? But it no longer remains the question of whether or not regularity is a fundamental property of all possible states of existence. And in fact it leaves open the question as to whether or not some kind of consciousness (perhaps very different from our own) could arise in what would appear to us as chaotic universes.

Re: Intuitionism

#112

Post by ndriscoll » Fri, Jul 14, 2023, 5:47 PM UTC

Der_Einzige wrote: post #30

The law of excluded middle always seemed like BS to me from the moment it was taught. It's wonderful that by removing it, proofs become "harder" to make, but consequently constructive and thus more rigoris. Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.

Even if you "believe" in LEM, I think it could be helpful to conceptualize proofs that use it as instead being proofs in the "Reader monad" over LEM. So instead of proving `A => B`, you prove `A => Reader[LEM, B] = A => (LEM => B)`. If you really examine the proof you're doing, probably you don't need LEM in its full power, but actually some specific `Not[Not[X]] => X` (or a handful of specific instances like that).

The neat thing about this is that in principle it could be tracked in a proof assistant with type inference. The ZIO framework for Scala has a super slick system for dependencies where e.g. an `RIO[Foo,A]` (an IO that requires a Foo and returns an A) and `A => RIO[Bar,B]` can compose to form an RIO[Foo&Bar, B] with the types inferred. So you could in principle have a proof system that lets you infer types like `A => Reader[LEM[Foo] & LEM[Bar], B]` (where LEM[T] = Not[Not[T]] => T), i.e. you get explicit types that show all of the instances of an LEM function you need to make your proof constructive, and since you're thinking of things as living in the Reader monad, you have the ergonomics of a proof that just assumes LEM.

Same idea could be used with AC or any other "controversial" axiom. So you don't need to "believe" them to use them, and if you do "believe" them, you can benefit from pretending you don't.

Re: Intuitionism

#113

Post by trabant00 » Fri, Jul 14, 2023, 5:53 PM UTC

simonh wrote: post #39

codeflo wrote: post #34

Earlier quoted context omitted.

I'm not sure what you mean. Presburger arithmetic is famously complete. What a system can't be is consistent, complete, and strong enough to perform a Godel encoding (which requires something multiplication-like). Drop any of the three requirements and it's possible. Inconsistent: trivial, from falsehood follows anything. Incomplete: Peano. Weak: Presburger.

The comment could be interpreted as meaning that such systems cannot be complete or consistent, I'm just pointing out they can be one or the other. As I understand it, it is possible to consistently prove and decide things in mathematics, just not everything. Godel proved limits to mathematics, not that mathematics doesn't work. That's all.

> Godel proved limits to mathematics, not that mathematics doesn't work

Nobody claimed it doesn't work. It clearly does. The question is if it's a fundamental property of the universe or just an useful but flowed human mental model.

Re: Intuitionism

#114

Post by mecsred » Fri, Jul 14, 2023, 6:03 PM UTC

mistercheph wrote: post #94

sunsetdive wrote: post #86

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.

The offer of 'objective reality' as the antithesis to solipsistic mental constructs is exactly the naivety that give both of these impotent families of epistemology any continued sway. Both bad, both wrong, and the crowd swings from one to the other, and at each arrival, anew recognizes the flaws of the mode and turns back.

Thank god a philosopher has arrived to tell us we're all wrong. Being so wise, you must have the correct answer for us. What's it going to be today "you're not smart enough to understand my genius solution" or "enlightenment can't be taught, only achieved"?

Re: Intuitionism

#115

Post by timeagain » Fri, Jul 14, 2023, 6:09 PM UTC

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…

To paraphrase Wittgenstein, is it more likely that we “discovered” chess or that we invented it? He suggests all of mathematics should be thought of in this way, as more and more complex ways of stating tautologies.

Re: Intuitionism

#116

Post by smokel » Fri, Jul 14, 2023, 6:37 PM UTC

Nasrudith wrote: post #106

danbruc wrote: post #16

Earlier quoted context omitted.

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". I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is usefu…

I suspect something opposite, that math comes before the universe because the universe needs math but math doesn't need the universe. Math only calls for internal consistency which radically does not depend upon the universe. It describe perfectly self-consistent realities that do not exist. Math could model say for instance a world of continuous matter as opposed to our mostly empty matter for instance. We have calc…

I beg to differ. Math typically requires abstract thought, symbols, and humans to produce and enjoy it. Especially the latter is quite a dependency.

We could reduce the requirements down to an implementation of a Turing machine, or something similar. (For the argument I simply ignore whether the machine is conscious -- that seems irrelevant in this context.)

That still requires some kind of discrete switch, which may seem fairly minimal to a human observer, but in reality consists of tens of thousands of atoms to operate. Atoms used to be simple, but turn out to be quite complex as well.

Representing, say, a circle in a fairly minimal system such as this would probably require the cooperation of millions of atoms.

Re: Intuitionism

#117

Post by staplers » Fri, Jul 14, 2023, 6:42 PM UTC

glcheetham wrote: post #87

rambojohnson wrote: post #83

Earlier quoted context omitted.

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

An illustration of AI disrupting society in indirect ways. If this comment were in fact written by human hands, your perception of it is lessened by the mere possibility it wasn't.

Re: Intuitionism

#118

Post by practal » Fri, Jul 14, 2023, 6:46 PM UTC

BSEdlMMldESB wrote: post #90

practal wrote: post #48

Earlier quoted context omitted.

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…

> math describes (fragments) of reality;

This is only possible if math itself is real. Note that I am not saying that a particular axiom system like Euclidean geometry has some sort of "real physical manifestation". No, what I am saying is that logical reasoning itself is real. And our reasoning about logical reasoning is certainly real as well, even if logical reasoning itself happens in very abstract form. Math itself might be viewed by some as just a game of symbols. But that doesn't change the fact that the game itself is real. Would it be otherwise, then math would be about as important as chess.

Re: Intuitionism

#119

Post by smokel » Fri, Jul 14, 2023, 6:46 PM UTC

pilgrim0 wrote: post #97

ttctciyf wrote: post #20

Earlier quoted context omitted.

> 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,…

> mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Yes. And such order exists within the observer, the mediators. We are measuring instruments. Scales are bound to the ruler and not to what’s being ruled. So mathematics represents order only in as much as there are people to validate it. Should all rulers be broken and forgotten, then there’s no measure at all. This is the sam…

Does your "principle which allows order" presume the existence of space and time, and more specifically the ordering of time? I would say that in a universe without time, "cause and effect" have little meaning.

I think I lost your train of thought when you say that numbers are qualities of Unit. Does your universe involve only "Unit", or are there other principles at play as well?

Re: Intuitionism

#120

Post by smokel » Fri, Jul 14, 2023, 7:01 PM UTC

zoogeny wrote: post #111

ttctciyf wrote: post #20

Earlier quoted context omitted.

> 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,…

> mustn't this in turn depend on some pre-existing regularity Consider the set of all things, including the ordered and disordered. Consider the operation of taking subsets of that set. Consider that conscious entities such as us only arise in ordered subsets (for some definition of ordered). Those conscious entities would see their proximal environment as ordered. They might assume that the only things in the set of…

I spent quite some time thinking along these lines, but then I realized that it is very unlikely to be the case in our universe.

If our world would be an "accidental" subset of the universe, with ordering, then how can it be that this ordering is so consistent? Would it not be more likely that we'd live in a world that has only some order, or varying order depending on one's position in space and time? For example, I would expect a lot more miracles to happen, but every physics experiment turns out to be extremely consistent.

This led me to believe that there is another process at play. My thought experiment now assumes that observation enforces a certain kind of consistency in the laws of physics. That is, the systems that we use to observe something, must by their very existence result in very consistent patterns in the chaos. What this precisely looks like and how it operates is left as an exercise to the reader.