Large cardinals: maths shaken by the 'unprovable'

[Rubystars]

It's interesting but it probably goes over my head too. I passed calculus a long time ago but I don't remember very much at all.

Edit: Yeah I was right, I tried looking at it and pardon the expression, but it's all "Greek" to me. 

[Yaakov Mendel]

Oh, look, there is even such a thing as finite calculus:

http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences

I'm sorry, I don't want to sound rude or aggressive, but the truth is you really don't know what you're talking about... Finite differences by no means call the mathematical concepts of infinity or continuity into question. Finite differences are a numerical method used mainly as a way to find approximate solutions to boundary value problems (i.e. partial differential equations associated with boundary conditions) on a grid. Actually, they apply to differentiable functions, and differentiability implies continuity which, in turn, implies infinity !

Likewise, "discrete" does not mean "finite". The set of the integers is discrete and it is not bounded. As I explained earlier in this thread, "discrete" refers to the notion of infinite countability.

[Confederate Kahanist]

Oh, look, there is even such a thing as finite calculus:

http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences

I'm sorry, I don't want to sound rude or aggressive, but the truth is you really don't know what you're talking about... Finite differences by no means call the mathematical concepts of infinity or continuity into question. Finite differences are a numerical method used mainly as a way to find approximate solutions to boundary value problems (i.e. partial differential equations associated with boundary conditions) on a grid. Actually, they apply to differentiable functions, and differentiability implies continuity which, in turn, implies infinity !

Likewise, "discrete" does not mean "finite". The set of the integers is discrete and it is not bounded. As I explained earlier in this thread, "discrete" refers to the notion of infinite countability.

I couldn't have said it better myself. 

[Masha]

Oh, look, there is even such a thing as finite calculus:

http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences

I'm sorry, I don't want to sound rude or aggressive, but the truth is you really don't know what you're talking about... Finite differences by no means call the mathematical concepts of infinity or continuity into question. Finite differences are a numerical method used mainly as a way to find approximate solutions to boundary value problems (i.e. partial differential equations associated with boundary conditions) on a grid. Actually, they apply to differentiable functions, and differentiability implies continuity which, in turn, implies infinity !

Likewise, "discrete" does not mean "finite". The set of the integers is discrete and it is not bounded. As I explained earlier in this thread, "discrete" refers to the notion of infinite countability.

I know very well that "discrete" is not "finite." However re-read the text I quoted above: "Continuity is in some ways associated with infinity and infinitesimal. Since calculus is concerned with continuous numbers and continuous functions, the subject must confront the ideas of infinity and infinitesimal." The two conceptes are associated through the kinds of problems they give birth to and the ways of solving them.

Now, let's take fractals that describe the mathimatics of the real world. Fractals are discrete and not integrable. Ideally, they are infinite. But in nature, fractal shapes are obviously finite. My question is whether it is possible to describe the geometry of the natural world without an appeal to infinities. To program a fractal, you don't need an infinity. You need to stop your iterations at some finite place. A lot of the problems today are solved with computer programs which don't use infinities. I am not convinced that for every method that uses infinitity we cannot find some alternative approximate calculation that gets away from infinities. After all, everything is approximate and statistical in the real world.

Saying "you don't know what you are talking about" is a sign of a feeble mind, as any kind of an appeal to authority. I may not know "what I am talking about" as concerns specific areas, but having been trained in philosophy, I understand things in general and their interconnection.

[Yaakov Mendel]

Saying "you don't know what you are talking about" is a sign of a feeble mind, as any kind of an appeal to authority. I may not know "what I am talking about" as concerns specific areas, but having been trained in philosophy, I understand things in general and their interconnection.

You keep boasting about your PHD in philosophy but that does not in the least guarantee that you understand mathematics. As for calling me a "feeble mind", thanks for the compliment, but no, calling a spade a spade does not make me a feeble mind.

[Zelhar]

I don't want to get between the fight here. But as to the point of whether we need infinity in the "real" world:

I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.

On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line. It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation. Also if we don't have continuum, then we have polynomials without roots etc. And also transcendental numbers like pi and e which seem to have real world implications in physics etc wouldn't exist. Although they are finite numbers, you can never really reach them, only approximate them. But if they don't exist in the real world, then how can you have a disk of radius exactly 1, which would then have an area of exactly pi, if you don't have exactly pi.

[Yaakov Mendel]

I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.

Is it the "world" that is finite and discrete or is it our perception ? It seems to me that when you refer to the amount of calculations we can do, you mean our perception.
There is also an ambiguity as to how you define "discrete". If you mean "separable", then many dimensions of reality are certainly not separable. Again, it is our biased perception that makes real objects look as if they were separated. Maybe this is so because we need this representation of reality for the purpose of practical action.
Topology is a great source of thoughts when it comes to the notion of “separability”.

On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line.

I think one must distinguish between various dimensions of reality. Ideal objects such as a straight line exhibit properties that are very different from those shared by material objects. When you turn to ideal objects such as mathematical concepts, infinity is everywhere. But material objects are essentially bounded. Mass, heat, speed, etc. don’t go to infinity in the physical world.

It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation.

The set of exact calculations, that do not imply any finite approximation, that can be done by a human mind through mathematical relations, is infinite !

[Zelhar]

I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.

Is it the "world" that is finite and discrete or is it our perception ? It seems to me that when you refer to the amount of calculations we can do, you mean our perception.
There is also an ambiguity as to how you define "discrete". If you mean "separable", then many dimensions of reality are certainly not separable. Again, it is our biased perception that makes real objects look as if they were separated. Maybe this is so because we need this representation of reality for the purpose of practical action.
Topology is a great source of thoughts when it comes to the notion of “separability”.

Our perception is certainly not finite, since we can perceive infinite things as we do in math. An of course in the pure mathematical sense infinity exists. But in the physical sense it might not exist. By "discrete" I mean a discrete topological space which means it is also separable. Why are you so sure that "many dimensions of reality are certainly not separable", do you mean space-time- how can we know if it is continuous or discrete ? We certainly are unable to prove that it is continuous because any measure we take will always have a limited accuracy. But if the space-time is discrete then it may be possible that one day we'd be able to prove it.

On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line.

I think one must distinguish between various dimensions of reality. Ideal objects such as a straight line exhibit properties that are very different from those shared by material objects. When you turn to ideal objects such as mathematical concepts, infinity is everywhere. But material objects are essentially bounded. Mass, heat, speed, etc. don’t go to infinity in the physical world.

It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation.

The set of exact calculations, that do not imply any finite approximation, that can be done by a human mind through mathematical relations, is infinite !

Well I agree with that. It just that bounded objects still appear to be continuous (or at least not separable) and so- being made out of an infinity of infinitesimal quantities.

[Masha]

What is your highest degree, yaakov mendel?

[Yaakov Mendel]

What is your highest degree, yaakov mendel?

Let's leave that to the PM mailbox. I don't think most JTFers care about my educational background.

[syyuge]

Imagine that when decimal system was not fully operative, the pi was considered nearest to 22/7.