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Intuitionism

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Re: Intuitionism

#41

Post by peteradio »

simonh wrote: post #38
peteradio wrote: post #35

Earlier quoted context omitted.

Highly consistent!? Not really. Some things are, but like OP just said, those are the ones we glom onto. But don't mistake some things for every things, there's a whole big wide world out there. We can hardly describe a ripple in a stream let alone why I've had the 5th argument in five weeks about the order I have to fix the kitchen with me wife, yet my intuition told me it was a comin.

So you consistently perceive streams, the world, have a body, a life, a wife, a kitchen with a flaw that has persisted over time, an order in to fix it. Also the world is consistent enough that you could predict that argument in advance. That sounds like an awful lot of consistency :)

Yes, I've already conceded that some things are consistent. Will you concede that we tend to glom onto them or will I interpret your reply where you once again glommed onto them as a concession? ;)

Re: Intuitionism

#42

Post by loicd »

simonh wrote: post #33
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…

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.

> Systems of mathematics cannot be both complete and consistent

No. They can't be at the same times complete, consistent, decidable and powerful enough to express arithmetic. You can do complete, consistent and decidable though.

Re: Intuitionism

#43

Post by BSEdlMMldESB »

constantcrying wrote: post #22
mg wrote: post #21

In the philosophy of mathematics, intuitionism is an approach where 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 Wouldn't that mean that not only mathematics is pure mental activity, but every thought? When we say (or think) "Joe and Sue went to the grocery", it seems inheren…

>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?

Re: Intuitionism

#44

Post by BSEdlMMldESB »

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?

if the system is sound, then (I think) by definition you cannot prove different (wrong) conclusions.

if you derive a different result, by soundness those would be equivalent ???

Re: Intuitionism

#45

Post by BSEdlMMldESB »

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…

> It's impossible for me to clearly find the number 1 (for example) anywhere.

what about that there is one of you?

Re: Intuitionism

#46

Post by dr_dshiv »

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

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 beyond human perceptual idiosyncrasy.

But I'm a deep Platonist/Pythagorean — so my bias is that "all is number" and the world is made of math. Math is real :)

Re: Intuitionism

#47

Post by haswell »

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?

My first foray into math involved counting with my fingers.

Re: Intuitionism

#48

Post by practal »

BSEdlMMldESB wrote: post #44
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?

if the system is sound, then (I think) by definition you cannot prove different (wrong) conclusions. if you derive a different result, by soundness those would be equivalent ???

It does not really matter if the system is sound or not, right? Although of course a sound one is far more interesting. Anyway, any way of justifying this is mathematical (and so would be the definition of soundness, if it was relevant here). If math is not real, then there is no justification.

Re: Intuitionism

#49

Post by bowsamic »

I agree with this but I would also take it further. I think that all science is purely the mental activity of humans. What is really happening is that we are probing the nature of our own mind, even in fields like physics (and I say this as a physicist myself). My justification for this is Kantian. We can't get away from the fact that our brain structures reality so that we can understand it, for example by organising things via our notions of space and time. We can't access the "noumena", i.e. the things as they truly are, independent of our perception of them. Therefore the study of physics, or indeed literally any activity, can only ever be transcendental and reflexive, rather than actually reaching out into the "objective world".

Re: Intuitionism

#50

Post by 112233 »

barrkel wrote: post #26
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…

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?

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