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Intuitionism

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Re: Intuitionism

#131

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.

I was initially surprised to read this because when I hear Intuitionism, I hear Intuitionist Logic. But IL doesn't have anything to do with denying objective reality; it can use facts on the way to proof. So I don't really know why Intuitionism is so much more adamant about denying constructive reality, or why it's thought to "give rise" to Intuitionist Logic, which at this point seems like a totally different thing. In other words, it's true that truth != proof, but that doesn't mean truth doesn't exist.

Re: Intuitionism

#132
The reality is that we can only have a sensible conversation about, or write sensibly about, concepts and facts within the context of our epistemic limit as a species.

You may have noticed that a lot of the cliché philosophical questions haven't really shifted for as long as people have been asking them. It isn't because there is no answer, but rather the question itself is null and void. It's answer is beyond our ability to find or compute because you would have remove yourself from the human frame of reference to find it.

The meaning of life? Doesn't make sense outside of the context of human life. In that context, it's whatever you want it to be or decide it is to you. Outside, well, you'd have to die to see if there's anything beyond, and as Spock points out in ST4, it would be impossible to discuss without a common frame of reference with someone who hasn't died.

~~~

To claim that math is a human invention or a noumenal language of nature is futile, the answer is not within our ability to determine. It is, for all intents and purposes, undefined. Unknowable.

That's not to say that the conversation is meaningless, as the logical positivists would claim, as this is also a leap too far.

Early Wittgenstein, big influence on the logical positivists, didn't try to claim that philosophy was meaningless, just that if you try to talk beyond the facts, you won't get anywhere.

Bradley aways summed it up nicely for me: "anyone ready to dispense with metaphysics is a brother metaphysician with a rival theory of his own"

In other words, to rule out metaphysics (defined as concepts and ideas that lack empirical data or basis in fact) is to make a metaphysical proposition, since by definition, there is no data to disprove the metaphysical propositions, just as much as there is no data to validate them.

Most metaphysical philosophies caught on because they can be presented rationally as a chain of propositions and conclusions, but they only add up within their own framework that is upheld by an unprovable assumption or set of assumptions.

Example, cogito ergo sum makes perfect sense, provided that you accept that the speaker is I, that the speaker thinks rather than simply utters, and that to be means anything at all. In the end, cogito ergo sum can be accepted intuitively within the framework of "let's not be sceptical of every thing" but what can you conclude from it?

"I think therefore I am" well, I is a thing that thinks

-> "I thinks, therefore it exists" I think presupposes I's existence

-> "I thinks"

That is a definition of I

I = thinks === I is a consciousness

which is a tautology. And that's why is makes sense, because it doesn't go anywhere than where it started, it's algebra.

It starts off as a = b where b = a and can be reduced to simply a

Math is privileged in that its algebraic statements can be used to model things in the world because it's numbers and functions of numbers, but ultimately the usefulness emerges from proving that a equals a very complicated statement that isn't obviously tautologically equal, or not equal to, a.

Math is a tool. It's a way of reasoning that comes bundled with reasonably standardised notation that enables boosted productivity compared to reasoning about the same problems in regular languages.

~~~

It's the same reason why I find "simulated reality" questions rather dull. If the world is a perfect simulation, we have no way to distinguish it from the real thing, and so to our frame of reference, there is no difference about which we can have a discussion that makes any sense.

If there are cracks in the simulation, sure, then you have a fact to talk about. But then the discussion isn't philosophical, it's practical: We've been brain-hacked, what do we do? and the conversation ends rather abruptly.

Re: Intuitionism

#133

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.

No, you're reading too much into it. The origin of the idea was a skepticism around seemingly paradoxical mathematical constructions, like uncountable infinities. Intuitionism eliminates some methods of proving that such things exist without needing to construct a proof of their existence.

And via Curry-Howard, any intuitionistic proof is also a computer program. Intuitionism thus unifies computation and mathematics in a very direct way, which has been extremely useful.

Re: Intuitionism

#134
post #120

Earlier quoted context omitted.

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…

> observation enforces a certain kind of consistency in the laws of physics. That sounds like a typical chicken-and-egg problem. > how can it be that this ordering is so consistent? One thing that I didn't explicitly state is that the universe only needs to seem ordered. You are making an assumption that isn't necessarily founded. 99.9% of the time we aren't checking on the ordered state of the universe. It is the ex…

I don’t have anything to add, but I just genuinely love your analogy. Totally helped me see the idea in a different way

Re: Intuitionism

#135
>mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality

In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.

Re: Intuitionism

#136

Earlier quoted context omitted.

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 its…

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 is ideal.

i suppose what it all is all about is the intersection between this reality and this ideality.

Re: Intuitionism

#137

>mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.

What is the opposite of an objective reality?

Re: Intuitionism

#138

There's something truly evil about psychologizing math. If anyone feels compelled by this you should probably start by learning about the many rock-solid antipsychologistic arguments out there that are at least 130 years old: https://plato.stanford.edu/entries/psychologism/#FreAntArg

Intuitionism has nothing to do with psychologizing math. The intuitionist is perfectly fine accepting that "the mind of the mathematician" is merely a metaphor for some system capable of mathematical constructions, and that the "mathematician" in this argument can be replaced by a computer performing the same steps. In fact, a lot of computer science, particularly programming language theory, depends on this very notion.

Re: Intuitionism

#139

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 true" question only affects the person thinking about it. The whole books are written that basically quibble about dictionary definitions of "fundamental principles", "objective reality", "mental activity" and other complex words.

(This is especially visible in discussion of "consciousness": there are tons of texts about it, and yet none of them matter in any practical way. The practical applications -- generative text models, NLP, neuroscience, etc.. -- just ignore all the philosophic cruft)

Re: Intuitionism

#140

>mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.

Gödel was a platonist btw.
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