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Intuitionism

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Re: Intuitionism

#21

Post by mg » 

    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 inherent to this statement that there are two distinct actors. Joe and Sue. But that is already math, isn't it? I have the feeling to avoid math, we would have to avoid "something" in the first place. As it already implies that "something" is in a category, consists of a collection of other things etc. So we could not talk about anything anymore.

Re: Intuitionism

#22

Post by constantcrying »

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.

Re: Intuitionism

#23

Post by mg »

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.

"pure mental activity" as opposed to "mental activity related to a reality outside of the mental activity".

Re: Intuitionism

#24

Post by constantcrying »

mg wrote: post #23
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.

"pure mental activity" as opposed to "mental activity related to a reality outside of the mental activity".

I don't see how you could distinguish those. Even the purest form mathematics involves things like rules and symbols and those have to exist in reality (certainly they are at least as real as any emotion).

Re: Intuitionism

#25

Post by trabant00 »

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 impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.

Re: Intuitionism

#26

Post by barrkel »

practal wrote: post #12
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 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.

Re: Intuitionism

#27

Post by simonh »

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…

I think both are descriptive. Mathematics is a descriptive language, and thoughts are descriptions of the world, ourselves, our intentions, etc or maybe the process of forming those descriptions.

Re: Intuitionism

#28

Post by simonh »

smokel wrote: post #2

I 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…

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.

Re: Intuitionism

#29

Post by practal »

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.

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?

Re: Intuitionism

#30

Post by Der_Einzige »

The law of excluded middle always seemed like BS to me from the moment it was taught. It's wonderful that by removing it, proofs become "harder" to make, but consequently constructive and thus more rigoris.

Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.

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