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Intuitionism

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Re: Intuitionism

#51

Post by bowsamic »

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 agree and this is actually a common problem in philosophical discussions. People struggle to differentiate between two kinds of objectivity: the "total" objectivity of knowledge that is completely context-free and unstructured, and the objective modulo a given set of assumptions (e.g., "the human brain", or at least, the human way of structuring and understanding reality). Mathematical truths are objective with respect to the latter kind of objectivity, but not the former.

Re: Intuitionism

#52

Post by bowsamic »

tgv wrote: post #17
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.

That doesn't really hold when you start measuring, unless you believe that your eyes can change the size of a measure tape depending on the subject.

> unless you believe that your eyes can change the size of a measure tape depending on the subject

Well, of course, they kind of can. There are drugs that make the world look like it's squashed, such as ketamine. The way that we perceive reality really is totally dependent on the properties of the observer. Of course we all, with our sober minds, assert that we are perceiving the ruler the "right way", but all this means it that we perceive the ruler in a way that most humans agree with. Jumping from that to "this is the way that the ruler looks for all possible subjects" is a leap of faith.

Re: Intuitionism

#53

Post by bowsamic »

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.

Yeah, this was the big step from Kant to Hegel, the realisation that the object is actually totally inside the subject and vice versa. Unfortunately, when the subject and object get totally mixed up in that way, the philosophy seems to become much more difficult and complicated. Kant's Transcendental Idealism is really useful and easy to understand, but if you want to go a step further into what you describe then it's like moving from Newtonian gravity to General Relativity. Literally everything becomes way more difficult.

Re: Intuitionism

#54

Post by constantcrying »

BSEdlMMldESB wrote: post #43
constantcrying wrote: post #22

Earlier quoted context omitted.

>Wouldn't that mean that not only mathematics is pure mental activity, but every thought? That statements seems very obviously true.

counterpoint: remove all blackboards and chalk, and paper and pencils. can you still do math?

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.

Re: Intuitionism

#55

Post by constantcrying »

Der_Einzige wrote: post #36
constantcrying wrote: post #32

Earlier quoted context omitted.

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.

Do you believe that any thing can be seperated into two part, which, if put together exactly the same as before become the initial part?

Re: Intuitionism

#56

Post by Tainnor »

constantcrying wrote: post #31
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…

There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?).

ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics.

You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.

Re: Intuitionism

#57

Post by skissane »

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.

> Inconsistent: trivial, from falsehood follows anything.

Only trivial if you accept the principle of explosion (ex falso quodlibet or ex contradictione quodlibet). If you reject it, you end up with paraconsistent logic, from which one can develop nontrivial inconsistent mathematics see https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

Re: Intuitionism

#58

Post by AnimalMuppet »

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…

"1" works just fine for things that are discrete. Take eggs, for example. I have one egg. If it's smaller than other eggs, I don't have 0.9 eggs; no, I have exactly one egg or, if you prefer, I have exactly one small egg, but still exactly one.

I have exactly one wife. If she gained weight, I would not then have 1.01 wives.

I have exactly one cat. If she had N kittens, I would then have exactly N+1 cats, not (1 * N/10) cats.

And so on.

Re: Intuitionism

#59

Post by skissane »

Tainnor wrote: post #56
constantcrying wrote: post #31

Earlier quoted context omitted.

>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…

There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?). ZFC is weird (especially choi…

Human thought is paraconsistent - relevance/relevant logic best models how implication works in natural language, and that is paraconsistent; I think the intelligibility of inconsistent fiction such as Graham Priest’s Sylvan’s Box [0] is also evidence of that. If one believes mathematics is ultimately grounded in human thought, and if human thought is ultimately paraconsistent, that suggests paraconsistent logic may be a better foundation for mathematics than classical logic. It also suggests that maybe we should seriously consider taking the inconsistency horn of Godel’s trilemma (incomplete or inconsistent or weak), given the paraconsistent rejection of the principle of explosion means that doing so is non-trivial. Inconsistent theories can be strong, complete and non-trivial.

[0] https://projecteuclid.org/journals/notre-dame-journal-of-for...

Re: Intuitionism

#60

Post by ttctciyf »

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…

> 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 follow a rule without a source of order or regularity in the cosmos? Wouldn't it be like trying to build a the Eiffel tower out of live slugs?

If you insist on an absence of order in the physical universe, the onus is on you to explain how regularity in human activity (required for mathematics of any kind) can be achieved without it.

This is not an argument for Platonic realism, BTW, or against intuitionism, roughly construed as the view that "mathematics is a creation of the mind" as per [1], or in your formulation that 'mathematical abstractions are not part of the physical universe' (if I understand what you're saying). You can perfectly well believe that mathematics is a mental construct and at the same time acknowledge that it's possible to observe regularities and order in the cosmos. If you want to insist that the regularity doesn't come from physical law, then I find it hard to see how you'll escape from some kind of Platonic belief in a non-physical realm that serves as the source of order :)

In your TV screen dots analogy, isn't it usually thought that patterns appear only because of the structured, generative activity of law-observing physical components, specifically the neurons comprising your grey matter?

1: https://plato.stanford.edu/entries/intuitionism/

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