|R| = |NxN|

[godelian]

The foundations of mathematics are philosophically problematic. This problem became increasingly understood over a century ago, and it will never get solved.

Something which doesn't exist cannot be problematic.

You can't solve a problem that you can't even define.

So, we take a set of beliefs and construct an abstract, Platonic world with it, which obviously does not exist physically. You are arguing that one of these beliefs should not be adopted, and that therefore, it cannot exist in the resulting Platonic world. As far as I am concerned, it is fine to adopt this belief and therefore it can exist in the resulting Platonic world.

When you write "something which doesn't exist", it sounds too much inspired by physicalism.

[Skepdick]

The foundations of mathematics are philosophically problematic. This problem became increasingly understood over a century ago, and it will never get solved.

Something which doesn't exist cannot be problematic.

You can't solve a problem that you can't even define.

So, we take a set of beliefs and construct an abstract, Platonic world with it, which obviously does not exist physically. You are arguing that one of these beliefs should not be adopted, and that therefore, it cannot exist in the resulting Platonic world. As far as I am concerned, it is fine to adopt this belief and therefore it can exist in the resulting Platonic world.

When you write "something which doesn't exist", it sounds too much inspired by physicalism.

Running for the hills of Platonism doesn't help.

The foundation doesn't exist even in the Platonic realm. So even asserting that there exists some kind of "problem" renders it incoherent.

[godelian]

Running for the hills of Platonism doesn't help.

The foundation doesn't exist even in the Platonic realm. So even asserting that there exists some kind of "problem" renders it incoherent.

The Platonic realm is constructed from beliefs only. You pick one belief that you do not like. Ok, in that case create another Platonic realm that does not use this belief in its construction.

[Skepdick]

Running for the hills of Platonism doesn't help.

The foundation doesn't exist even in the Platonic realm. So even asserting that there exists some kind of "problem" renders it incoherent.

The Platonic realm is constructed from beliefs only. You pick one belief that you do not like. Ok, in that case create another Platonic realm that does not use this belief in its construction.

Uhuh. Constructed FROM sometthing.... The resources FOR constructing the Platonic realm are not themselves platonic.

[godelian]

Running for the hills of Platonism doesn't help.

The foundation doesn't exist even in the Platonic realm. So even asserting that there exists some kind of "problem" renders it incoherent.

The Platonic realm is constructed from beliefs only. You pick one belief that you do not like. Ok, in that case create another Platonic realm that does not use this belief in its construction.

Uhuh. Constructed FROM sometthing.... The resources FOR constructing the Platonic realm are not themselves platonic.

They are abstract beliefs that are just as platonic as the world created.

[Skepdick]

They are abstract beliefs that are just as platonic as the world created.

So there are created and uncreated Platonic objects?

What kind of Platonic object are problems?

[godelian]

They are abstract beliefs that are just as platonic as the world created.

So there are created and uncreated Platonic objects?

What kind of Platonic object are problems?

The completely inducted set of natural numbers is a Platonic abstraction, but so is the set {1,2,5}. I don't see why the one would not exist while the other would.

[Skepdick]

They are abstract beliefs that are just as platonic as the world created.

So there are created and uncreated Platonic objects?

What kind of Platonic object are problems?

The completely inducted set of natural numbers is a Platonic abstraction, but so is the set {1,2,5}. I don't see why the one would not exist while the other would.

What does this have to do with my question?

Also...

Code: Select all

In [1]: set({1,2,3})
Out[1]: {1, 2, 3}

In [2]: set(range(1,10))
Out[2]: {1, 2, 3, 4, 5, 6, 7, 8, 9}

[godelian]

So there are created and uncreated Platonic objects?

What kind of Platonic object are problems?

The completely inducted set of natural numbers is a Platonic abstraction, but so is the set {1,2,5}. I don't see why the one would not exist while the other would.

What does this have to do with my question?

Also...

Code: Select all

In [1]: set({1,2,3})
Out[1]: {1, 2, 3}

In [2]: set(range(1,10))
Out[2]: {1, 2, 3, 4, 5, 6, 7, 8, 9}

Platonic existence as an abstraction is different from physical existence as a phenomenon.

Abstractions exist because you decide that they do. You achieve this by explicitly pronouncing them in language. Physical phenomena, on the other hand, exist because you can observe them in physical reality.

The axiom of infinity creates the abstraction of the completely inducted set of the natural numbers by decree. It is no different from any other abstraction, because in the end, they all get created by axiomatic decree.

Your criticism on the axiom of infinity ultimately amounts to the very rejection of the axiomatic nature of mathematics.

[Skepdick]

The completely inducted set of natural numbers is a Platonic abstraction, but so is the set {1,2,5}. I don't see why the one would not exist while the other would.

What does this have to do with my question?

Also...

Code: Select all

In [1]: set({1,2,3})
Out[1]: {1, 2, 3}

In [2]: set(range(1,10))
Out[2]: {1, 2, 3, 4, 5, 6, 7, 8, 9}

Platonic existence as an abstraction is different from physical existence as a phenomenon.

I think you got that backwards. For somebody who preaches a syntactic view of Mathematics, you sure are appealing to semantics.

Abstractions exist because you decide that they do. You achieve this by explicitly pronouncing them in language. Physical phenomena, on the other hand, exist because you can observe them in physical reality.

I can observe my abstractions too... they are all in my head. Same place all all other thoughts.

The axiom of infinity creates the abstraction of the completely inducted set of the natural numbers by decree. It is no different from any other abstraction, because in the end, they all get created by axiomatic decree.

OK... so I accept the axiom which negates infinity.

Your criticism on the axiom of infinity ultimately amounts to the very rejection of the axiomatic nature of mathematics.

Bingo. Axiomatics is bullshit.

What we deal with in programming is equational reasoning. Dynamic systems and systems of equations.

It's about expressing the control-flow, not the data-flow. Operational (how things execute), not denotational semantics (what things mean). Ultimately, that's what the completeness property is.

Can your system of control flow equations completely describe the execution behaviors you want?
Are the operational semantics complete enough to express all the control patterns needed?
Does the system have sufficient expressiveness to model all the ways control needs to flow?

Does your programming system have all the control flow mechanisms needed to express the computations you want to perform?

[godelian]

OK... so I accept the axiom which negates infinity.

Yes, that is fine. ZF-inf does exactly that.

Your criticism on the axiom of infinity ultimately amounts to the very rejection of the axiomatic nature of mathematics.

Bingo. Axiomatics is bullshit.

That is a courageous point of view. For me, it would not work, because that means that pretty much the entire body of literature on mathematics would become unusable. I don't want to go it alone.

In software engineering, that would amount to adopting a view that rejects the use of every compiler and every scripting engine. In that case, what do you use to make programs executable?

[Skepdick]

That is a courageous point of view. For me, it would not work, because that means that pretty much the entire body of literature on mathematics would become unusable. I don't want to go it alone.

In software engineering, that would amount to adopting a view that rejects the use of every compiler and every scripting engine. In that case, what do you use to make programs executable?

That's literaly what I am talking about. Different execution engines. Programming languages. Models of computation. Potato/potatoh.

Expressing control flow. Operational semantics.

Turing completeness is the default - no axioms. Universal computational system. Does nothing. Now write some transformations.

Construct an algebra.

No axioms needed beyond the basic rules of how operations transform states.

Now show me this "axiom of infinity" and explain to me how I go about making the computer pretend it has DONE an infinite number of iterations.
How many actual operations does it take to pretend you've done infinitely many of them?