Intuitionism
en.wikipedia.org
Intuitionism
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Re: Intuitionism
#2I 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 as humans could be limited to the hyperplane of our own observable patterns, which fools us into thinking there is some inherent order -- which there isn't. This leaves us with "why is there chaos rather than nothing", so I doubt it's of any help :)
Great ideas to ponder, but rather hard to reason about.
(Edit: To avoid thinking that I'm a crackpot, with "capability of understanding", I am referring to the physical processes that lead to the existence and dynamics of neurons, not to the platonic world of ideas on top of that. If someone could point out how unoriginal or nonsensical this idea is, it would save me from writing a blog post about it.)
Re: Intuitionism
#3The 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 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.
A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.
Re: Intuitionism
#4Because 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.
Re: Intuitionism
#5The 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.
Re: Intuitionism
#6The 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…
But the subsequent paragraph talks about the truth of proofs themselves as being subjective.
This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p).
Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.
Re: Intuitionism
#7[deleted]
Re: Intuitionism
#8Because 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.
This is the transcendental idealism of Kant. He makes a distinction between the noumena (things in themselves beyond human cognition) and phenomena (things as they appear to us through our senses). Our mind constructs "transcendental objects" which are merely abstract ideas based on the appearances.
Re: Intuitionism
#9I 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…
It's not chaos, it's infinitely ordered order most would rather die than accept.
Re: Intuitionism
#10The 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…
>mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems.
This is essentially the standard mathematical approach developed during the early last century. From a few basic axioms (which are not really justifiable) new statements are proven and structures are built up. All notions of truth and provability are relative to that system. In standard ZFC (those are the standard axioms) mathematics "1+1=2" is, like all other statements, a statement about sets. The statement is true by the definitions of 1,2, "=" and "+". In an alternative system with different axioms or definitions the statement is false.
This is not the view of intuitionists though. For them the symbols 1,2 or + aren't formalized objects (e.g. sets in this case). They are just symbols, which transfer some (hopefully) shared meaning to another person, who then (potentially with some addition arguments) might also accept the truth of that statement.
In the former the question of "truth" is fully formal there can be no "interpretation" in any meaningful sense. Intuitionism places mathematics fully inside the mind of the mathematician and "truth" can only be found there.
>Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality?
In formalized mathematics "truth" is fully formal. There can not be any "external" truth derived from it, as any such statement is non-sensical.