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Intuitionism

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Re: Intuitionism

#61

Post by barrkel »

112233 wrote: post #50
barrkel wrote: post #26

Earlier quoted context omitted.

I think it's a system of symbols and rules. By mechanically provable, I mean that given axioms (assumptions) and rules, you can devise a machine (i.e. something which follows rules, with no independent thinking or homunculus) which generates statements which follow from the axioms and rules, and this is what "true" means in the system.

Wow, you excavated an ocean to cover a puddle. how this: "something which follows rules, with no independent thinking or homunculus" can be easier to prove and reason than the initial rules? For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?

The reason I brought up a machine explain something as being mechanical is to clarify that it doesn't require intuition.

We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.

Re: Intuitionism

#62

Post by barrkel »

practal wrote: post #29
barrkel wrote: post #26

Earlier quoted context omitted.

I think it's a system of symbols and rules. By mechanically provable, I mean that given axioms (assumptions) and rules, you can devise a machine (i.e. something which follows rules, with no independent thinking or homunculus) which generates statements which follow from the axioms and rules, and this is what "true" means in the system.

So would you say that your system of symbols and rules is real? Could it be that we both use the same system of symbols and rules, with the same assumptions, but derive different conclusions? If not, why not?

It has no substance other than its representations; there's nothing of it you can touch which is physical. At best, there is a correspondence between the physical and the system.

Re: Intuitionism

#63

Post by barrkel »

mjw1007 wrote: post #5
barrkel wrote: post #3

The article seems poorly written, or at least self-contradictory in a way that makes me uncertain what it means. The introduction talks about the intuitionism framing mathematics as a human construction, in opposition to an objective reality. But the subsequent paragraph talks about the truth of proofs themselves as being subjective. It seems to me that these are two different claims. I think mathematics is a human c…

I think the Stanford Encyclopedia of Philosophy article is better. https://plato.stanford.edu/entries/intuitionism/

Yeah, that's much better.

Re: Intuitionism

#64

Post by RubyRidgeRandy »

simonh wrote: post #14
RubyRidgeRandy wrote: post #4

Because humans never experience anything in and of itself, but only the output of the interaction between sensory data and a brain, literally everything is purely the result of human mental activity.

I would agree that everything we experience is a model of the world that we construct from sense data, interpreted by our sensory systems and cognitive faculties. Donald Hoffman is good on this and worth looking up, although I disagree with some of his conclusions. That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce th…

I think there is a hard limit to what we know and what we can assume to know based of this point and in logic by the Münchhausen trilemma. It's interesting to think of the source of sense data as persistent or consistent when it could just be that our sense organs reduce varied data into persistent experience.

When we look at a tree, it could very well be that the source of the tree is very much like the tree we experience, but it could also be wildly different. When we see a tree in a video game, we know there is no real source tree just like it, just ones and zeroes. I disagree that science fixes this problem. Tools are still just measuring the physical world. For example, if you used a tool to measure some aspect of the tree, you are still measuring the representation of the tree in this world. If I use the video game analogy again, my point is that you wouldn't be able to see true underlying 'source code' of the game tree by looking at it in the game.

Re: Intuitionism

#65

Post by RubyRidgeRandy »

circlefavshape wrote: post #18
RubyRidgeRandy wrote: post #4

Because humans never experience anything in and of itself, but only the output of the interaction between sensory data and a brain, literally everything is purely the result of human mental activity.

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.

Re: Intuitionism

#66

Post by practal »

barrkel wrote: post #62
practal wrote: post #29

Earlier quoted context omitted.

So would you say that your system of symbols and rules is real? Could it be that we both use the same system of symbols and rules, with the same assumptions, but derive different conclusions? If not, why not?

It has no substance other than its representations; there's nothing of it you can touch which is physical. At best, there is a correspondence between the physical and the system.

I agree with you here, at least in the sense that the system is definitely not physical. You didn't answer my question, though. Is it real?

Re: Intuitionism

#67

Post by pixl97 »

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…

>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...

Re: Intuitionism

#68

Post by klik99 »

ttctciyf wrote: post #60
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…

> 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 infinity, pure chaos does not preclude long pockets of what looks like order. What we consider fundamental rules could very well be local phenomenon, which we are a product of.

Of course this purely theoretical - all I’m saying is that intuitism could be true while also math being useful to predict things right now. We could also only exist for an instant and all our memories just construct, but that’s not very useful. It’s more useful to believe in scientific method because what’s repeatable is provable, whereas chaos is by its nature unprovable - which doesn’t make it impossible.

Re: Intuitionism

#69

Post by denotational »

constantcrying wrote: post #15
practal wrote: post #12

Earlier quoted context omitted.

> I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable. Mathematics is a human construction, but it certainly has a real substance. What d…

>What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not? Different Axioms lead to different provable statements. Believing in standard mathematics basically means that you can not believe in absolute truth. Unless you also believe that some guys a hundred years ago figured the sole and completely perfect rules which totally correspond to reality.

> Believing in standard mathematics basically means that you can not believe in absolute truth.

I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this as "standard"; it seems to me to be a very Formalist stance.

I'd argue that most mathenaticians consider themselves Platonists, and believe that the mathematical objects they are describing are real enough to form some kind of metamathematical "standard model", and "absolute truth" can be defined in the model-theoretic sense relative to this standard model, even if this is somewhat unavoidably handwavy.

[1] : Even if you do think this, "truth" is generally used by logicians in the model-theoretic sense of "truth in some specific model/structure compatible with the language".

Re: Intuitionism

#70

Post by antonvs »

dr_dshiv wrote: post #46
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 Aren’t all basic particles defined by the fact that they are exactly the same? And countable things rely on some difference, such as different spatial locations — or else they wouldn’t be countable, they’d just be the same. While two neutron stars are distinguishable they are also classifiable as a real type of star in a manner that seems to go be…

If the world is made of math, do new physical objects pop into being whenever a mathematician writes down or thinks of a structure or a proof? Or does only some math get to become physical?

In general, your view seems like a definitional issue to me. If you want to call what the world is made of “math”, then what you and I mean by math are two different things, and using the same word to describe them only leads to confusion.

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