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

31–40 of 179 posts

Re: Intuitionism

#31

Post by constantcrying »

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…

>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 that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.

Re: Intuitionism

#32

Post by constantcrying »

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.

It directly leads to the continuum being inseperable.

If you believe that "now" is real, you believe in LEM.

Re: Intuitionism

#33

Post by simonh »

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…

That's not quite correct. Systems of mathematics cannot be both complete and consistent, but incomplete systems of mathematics can be consistent. For example Presburger arithmetic is provably consistent. There are limits to consistency for sure, but that doesn't mean there's no such thing as mathematical consistency.

Re: Intuitionism

#34

Post by codeflo »

simonh wrote: post #33
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…

That's not quite correct. Systems of mathematics cannot be both complete and consistent, but incomplete systems of mathematics can be consistent. For example Presburger arithmetic is provably consistent. There are limits to consistency for sure, but that doesn't mean there's no such thing as mathematical consistency.

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.

Re: Intuitionism

#35

Post by peteradio »

simonh wrote: post #28
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…

The physical world is a persistent system that exhibits highly consistent behaviour. Because the behaviour is consistent we can describe it in a highly consistent formal language, mathematics. What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.

Highly consistent!? Not really. Some things are, but like OP just said, those are the ones we glom onto. But don't mistake some things for every things, there's a whole big wide world out there. We can hardly describe a ripple in a stream let alone why I've had the 5th argument in five weeks about the order I have to fix the kitchen with me wife, yet my intuition told me it was a comin.

Re: Intuitionism

#36

Post by Der_Einzige »

constantcrying wrote: post #32
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.

It directly leads to the continuum being inseperable. If you believe that "now" is real, you believe in LEM.

Well, I'm an eternalist who remembers the basics of the Relativity of simultaneity, so I reject that "now is real" as such.

Re: Intuitionism

#37

Post by smokel »

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,…

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 complex amalgam of matter comes together [1] to realize a discrete switch. Using these discrete switches, organisms build memory and abstraction mechanisms, start counting, and do mathematics.

So my thesis is that things that are considered to be "basic" by most, such as logic or mathematics, are in fact quite specific systems built on top of chaos. From there, it remains to be proven that all physical laws that we observe are no more than projections of the chaos onto such a system.

Perhaps a metaphor that helps to take on my perspective, is to look at a screen filled with random noise, and then observe some patterns in there. Now replace the screen with an infinite dimensional set of chaos, and then observe a pattern that is our universe. With the added twist that we are part of this chaos, and observing the pattern, possibly in the form of physical laws.

Of course, there are many problems with this theory. What does the chaos reside in? How can there be discernible parts in the chaos? Is time an emergent property inside the chaos? How can it be that the patterns that we observe are so consistent?

However, to me this theory seems more fruitful than merely accepting that we cannot say anything about things that science cannot observe, or that some deity created all this.

[1] With "comes together" I do not refer to a dynamic process, but to the accidental occurrence of stuff in such a shape or form. Obviously, my ridiculous theory asserts a chaos chock full of dimensions, where time and space are but supporting actors.

Re: Intuitionism

#38

Post by simonh »

peteradio wrote: post #35
simonh wrote: post #28

Earlier quoted context omitted.

The physical world is a persistent system that exhibits highly consistent behaviour. Because the behaviour is consistent we can describe it in a highly consistent formal language, mathematics. What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.

Highly consistent!? Not really. Some things are, but like OP just said, those are the ones we glom onto. But don't mistake some things for every things, there's a whole big wide world out there. We can hardly describe a ripple in a stream let alone why I've had the 5th argument in five weeks about the order I have to fix the kitchen with me wife, yet my intuition told me it was a comin.

So you consistently perceive streams, the world, have a body, a life, a wife, a kitchen with a flaw that has persisted over time, an order in to fix it. Also the world is consistent enough that you could predict that argument in advance. That sounds like an awful lot of consistency :)

Re: Intuitionism

#39

Post by simonh »

codeflo wrote: post #34
simonh wrote: post #33

Earlier quoted context omitted.

That's not quite correct. Systems of mathematics cannot be both complete and consistent, but incomplete systems of mathematics can be consistent. For example Presburger arithmetic is provably consistent. There are limits to consistency for sure, but that doesn't mean there's no such thing as mathematical consistency.

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.

Re: Intuitionism

#40

Post by narag »

smokel wrote: post #37
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,…

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…

I've seen statistics proposed as the force that makes reality, that would be fundamentally random, coherent. But statistics laws are themselves very strong when numbers get big.
Post reply on HN