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Intuitionism

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Re: Intuitionism

#151

Post by darkclouds »

darkclouds wrote: post #72
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.

> The physical world is a persistent system that exhibits highly consistent behaviour. Thats because chemicals shape/influence our personalities and emotions. You'll see the same consistent behaviours in animals as you do in humans, when given the same chemicals.

Dont know why this got down voted.

https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RS01....

Your mobile phone can be used as a weapon against you, triggering calcium waves in the body which could be harmful to your liberty and your life.

I know because I've had it done to me!

Re: Intuitionism

#152

Post by antonvs »

dr_dshiv wrote: post #142
antonvs wrote: post #127

Earlier quoted context omitted.

Again, definitional issues are critical here. I can make up all sorts of math that isn't really "discovered", it's more "invented". Most of it won't correspond usefully to the physical universe. See e.g.: https://plato.stanford.edu/entries/formalism-mathematics/ : > "mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitmen…

Take the brevity as a lack of hubris. The notion that “the world is made of math” is one of the oldest and most influential ideas of all time. If you find Pythagoras, Plato and Newton a little incoherent, that’s not unusual. But the onus doesn’t lie with them (or me). In any case, I remain interested in your ideas!

None of Pythagoras, Plato, or Newton claimed that the “world is made of math”. Also, Aristotle’s philosophy of mathematics is considered an alternative to Plato’s, so trying to seek solace in both at the same time seems inconsistent.

Plato described math as a realm distinct from both the physical world and the world of consciousness. This doesn’t support the idea the “the world is made of math”.

Newton described the world as operating in accordance with the rules of math, but that’s not the same as being “made of math.” Plato’s view is compatible with this: what Frege described as the “third realm”, the realm of abstract objects, can have a relationship with the physical realm without requiring that the latter be “made of” the former.

Aristotle explicitly distinguished between physics and mathematics, saying in his Metaphysics that physics is concerned with things that change, whereas mathematics encompasses things that are eternal, do not change, and are not substances. So Aristotle seems to explicitly reject your view.

As such, I don’t accept your claim that your position is “one of the oldest and most influential ideas of all time.”

> But the onus doesn’t lie with them (or me).

If you make a claim, the onus certainly lies on you to support that claim.

Re: Intuitionism

#153

Post by constantcrying »

BSEdlMMldESB wrote: post #96
constantcrying wrote: post #54

Earlier quoted context omitted.

Sure, it just takes a lot of remembering. But the point is that still that isn't "pure" mental activity. I am still imagining symbols and rules, those exist in reality at least as much as emotions do.

hmm, but imagining rules and symbols IS a pure mental activity, isn't it???

They are very similar to language, which cettainly is part of reality.

Re: Intuitionism

#154

Post by constantcrying »

denotational wrote: post #69
constantcrying wrote: post #15

Earlier quoted context omitted.

>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…

>but I don't agree with the charactedisation of this as "standard"

I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival theory against.

Re: Intuitionism

#155

Post by constantcrying »

markisus wrote: post #73
constantcrying wrote: post #15

Earlier quoted context omitted.

>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.

Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems. I forgot who said it but if we were to find a contradiction using Peano’s axioms, we would say that the axioms were wrong, rather than arithmetic itself. Even your comment references “perfect rules which totally correspond to reality” which seems to be another way to say “absolute truth”.

>Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems.

I am certain about that, but it does not make my statement less true. I actually think that very few people believe in ZFC as either a formalist absolute or as an arbitrary set of rules. I think the most common view is that it enables other theories, that those mathematicians actually care about. The moment those theories rely directly on axioms things get difficult. I think the following quote describes quite well the state of ZFC: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

Re: Intuitionism

#156

Post by Tainnor »

theamk wrote: post #139
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 started being much happier about philosophy in general once I realized a very important truth: "unlike hard sciences, philosophy only affects philosophers" For example the toplevel article says things like "independent existence in an objective reality".. this sounds important, right? Perhaps depending on the truth of the intuitionism, we might discover some facts about real world? Nope. The whole "is intuitionism…

> "unlike hard sciences, philosophy only affects philosophers"

Studying history will very quickly dissuade you from this notion, over and over again. For the perhaps most striking example, consider how different the 20th century would have looked like if a certain philosopher named Marx hadn't tried to marry the works of another philosopher, Hegel, with economic theory.

Re: Intuitionism

#157

Post by mensetmanusman »

ineedasername wrote: post #144
mensetmanusman wrote: post #137

Earlier quoted context omitted.

What is the opposite of an objective reality?

I don’t know, but an objective reality could exist without it being fully accessible to us or directly corresponding to our mathematical systems. I guess the opposite would be something like solipsism though.

Solipsism assumes the objective reality of the self.

Re: Intuitionism

#158

Post by ineedasername »

mensetmanusman wrote: post #157
ineedasername wrote: post #144

Earlier quoted context omitted.

I don’t know, but an objective reality could exist without it being fully accessible to us or directly corresponding to our mathematical systems. I guess the opposite would be something like solipsism though.

Solipsism assumes the objective reality of the self.

That would fit though, wouldn’t it? Objective reality == exists independently of the self. Solipsism == only the self is a given. At the very least it’s opposite-ish. And if not, then I simply don’t have a better answer, but it wasn’t my question to begin with.

Re: Intuitionism

#159

Post by [deleted] »

practal wrote: post #149
BSEdlMMldESB wrote: post #136

Earlier quoted context omitted.

I like to draw a distinction between real and ideal. I insist that math is ideal. it models reality ideally. this distinction is important because otherwise we mix together something, and the ideas and concepts (e.g. symbols and rules) we use to describe and model said something. the game is not real. people playing the game are real, the game getting played is real. the game on its own as may be described in symbols…

You can say that a certain axiom system models a certain part of reality in an ideal way. But whatever is ideal, is also real, because otherwise there is nothing that could model anything. So your intersection of reality and ideality is just ideality itself.

[deleted]

Re: Intuitionism

#160

Post by mecsred »

cubefox wrote: post #146
mecsred wrote: post #114

Earlier quoted context omitted.

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"?

Yeah. There are some approaches in philosophy of mathematics which try to avoid platonism (the view that mathematical objects have a mind independent existence) but while also retaining classical logic. That's not easily done though. (Currently popular is "structuralism", but this theory has its own problems.)

I do appreciate you providing some relevant information instead of just telling everyone else they're wrong. You still haven't put any of your own ideas on the line though. Is there an approach you believe is correct, and why?

Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature, they'd be scientific questions if they did. The goal is to massage the question into an answerable one if you can. It's not really possible to have a "wrong" answer if the question is unknowable or malformed.

I could have made the original post with less melodrama but I like to snub philosophers when they swoop in like they have answers to these questions. Either it's a matter of science or you don't have answers, just perspectives.

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