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Intuitionism

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Re: Intuitionism

#161

Post by theamk »

nextaccountic wrote: post #143
theamk wrote: post #139

Earlier quoted context omitted.

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…

Intuitionism has had a great impact on computer science and the design of many programming languages. It's now widely known that an intuitionistic proof has computational content (through the curry-howard correspondence) Those things have a real world impact

I have no doubt it had an impact on Curry personally, but it does not it has direct impact to on the the computer science. The Curry–Howard correspondence is formulated entirely in mathematical terms, and no philosophy is required to understand and use it.

For example, Robert Goddard, who invented first liquid-fueled rocket, became interested in space when he read Wells' "The War of the Worlds". So did Wells' work had an impact on rocket science? Certainly. Does it mean you should read "The War of the Worlds" if you want to build rockets? Probably not.

Re: Intuitionism

#162

Post by theamk »

Tainnor wrote: post #156
theamk wrote: post #139

Earlier quoted context omitted.

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…

> "unlike hard sciences, philosophy only affects philosophers" Studying history will very quickly dissuade you from this notion, over and over again. For the perhaps most striking example, consider how different the 20th century would have looked like if a certain philosopher named Marx hadn't tried to marry the works of another philosopher, Hegel, with economic theory.

And yet, it was not Marx or Hegel who took over the country and started campaign on Red Terror. And Lenin did not need to cite Das Kapital to start Russian Civil War. (I have no doubt he was inspired by it though)

(It's an interesting thought experiment what would have happened if there were no Karl Marx. February Revolution would have happened anyway.. but would bolsheviks still exist? Would Lenin still do all the things he did anyway, or would he become a civil servant instead?)

And for the most striking example, consider how different the 20th century would have looked if Academy of Fine Arts Vienna would have accepted a certain applicant from Linz. Does this mean fine arts educational institutions are very important in the political life?

Re: Intuitionism

#163

Post by kazinator »

> To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed

But, like, just in someone's mind ... to keep the truth of that existence purely subjective.

Re: Intuitionism

#164

Post by tunesmith »

cubefox wrote: post #145
tunesmith wrote: post #131

Earlier quoted context omitted.

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.…

If there is an independent source of truth (external reality), then classical logic makes sense and intuitionistic logic doesn't. But intuitionists say mathematics, unlike the physical would, doesn't have such an independent reality. There is no platonic mathematical reality apart from explicit mathematical construction. Then classical logic is inappropriate and intuitionistic logic has to be used for mathematics.

So then what kind of logic is the following:

"Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.

And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".

This causes "Socrates is mortal" to have a truth value of "False", which doesn't make sense unless you consider it "proof" instead of "truth": it used to be proven (by the premises) that Socrates is mortal, but now it is "false that it is proven". It might still be true, but it's not proven.

This is closer to intuitionist logic than classical logic, but it still relies on facts like "Socrates is male".

Re: Intuitionism

#165

Post by denotational »

constantcrying wrote: post #154
denotational wrote: post #69

Earlier quoted context omitted.

> Believing in standard mathematics basically means that you can not believe in absolute truth. I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this…

>but I don't agree with the charactedisation of this as "standard" I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival th…

Working, or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth.

As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.

I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth.

I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system.

You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.

Re: Intuitionism

#166

Post by dr_dshiv »

antonvs wrote: post #152
dr_dshiv wrote: post #142

Earlier quoted context omitted.

Take the brevity as a lack of hubris. The notion that “the world is made of math” is one of the oldest and most influential ideas of all time. If you find Pythagoras, Plato and Newton a little incoherent, that’s not unusual. But the onus doesn’t lie with them (or me). In any case, I remain interested in your ideas!

None of Pythagoras, Plato, or Newton claimed that the “world is made of math”. Also, Aristotle’s philosophy of mathematics is considered an alternative to Plato’s, so trying to seek solace in both at the same time seems inconsistent. Plato described math as a realm distinct from both the physical world and the world of consciousness. This doesn’t support the idea the “the world is made of math”. Newton described the…

Pythagoras said “all is number.” So, where some claimed that “fire” was the primary constituent of all things and others “earth,” Pythagoreanism held that numbers were the underlying principle. Do you accept this claim?

Newton was a Pythagorean. He even attributed the inverse square law to Pythagoras.

Plato is always hard to pin down, but he describes the immaterial world as crafted by number, prior to the material world.

Re: Intuitionism

#167

Post by constantcrying »

denotational wrote: post #165
constantcrying wrote: post #154

Earlier quoted context omitted.

>but I don't agree with the charactedisation of this as "standard" I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival th…

Working , or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth. As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system. I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth. I would hazard a guess that most (all?) working resear…

>Working, or at least claiming to work, in ZFC is fairly standard

Which is why I called it "standard".

> I don’t think we can say that ZFC is the standard basis of mathematical truth

You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards.

>As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.

Which I agree with.

Re: Intuitionism

#168

Post by denotational »

constantcrying wrote: post #167
denotational wrote: post #165

Earlier quoted context omitted.

Working , or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth. As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system. I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth. I would hazard a guess that most (all?) working resear…

>Working, or at least claiming to work, in ZFC is fairly standard Which is why I called it "standard". > I don’t think we can say that ZFC is the standard basis of mathematical truth You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards . >As a sibling comment mentioned, most mathematicians have a sense of trut…

You said:

> truth being relative only to ZFC is the standard mathematical foundation

I think this is at odds with:

> most mathematicians have a sense of truth that is not bound to any axiom system

Re.:

> You literally just said that working in ZFC is "standard"

Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.

Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.

Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”

Re: Intuitionism

#169

Post by constantcrying »

denotational wrote: post #168
constantcrying wrote: post #167

Earlier quoted context omitted.

>Working, or at least claiming to work, in ZFC is fairly standard Which is why I called it "standard". > I don’t think we can say that ZFC is the standard basis of mathematical truth You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards . >As a sibling comment mentioned, most mathematicians have a sense of trut…

You said: > truth being relative only to ZFC is the standard mathematical foundation I think this is at odds with: > most mathematicians have a sense of truth that is not bound to any axiom system Re.: > You literally just said that working in ZFC is "standard" Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite. Deciding to adopt a particular…

>I think this is at odds with:

I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.

>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”

All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.

I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.

My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.

The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.

Re: Intuitionism

#170

Post by mensetmanusman »

ineedasername wrote: post #158
mensetmanusman wrote: post #157

Earlier quoted context omitted.

Solipsism assumes the objective reality of the self.

That would fit though, wouldn’t it? Objective reality == exists independently of the self. Solipsism == only the self is a given. At the very least it’s opposite-ish. And if not, then I simply don’t have a better answer, but it wasn’t my question to begin with.

It seems that it builds towards your questions though, and we would be better off if we could come to some agreement on objective reality.

Although I understand it’s unlikely philosophers will ever agree on what reality is.

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