Does infinity exist?

[beowulf]

Put this one here since it is a science question but also one of philosophy. Looks like an interesting documentary and I must admit have spent time in my life contemplating this idea of infinity. Does the universe really go on forever. Is the only real thing that is infinite is being dead? Do numbers go on forever?

Click to read more...

 

[JMCx4]

 

[The Crypto Guy]

My brain hurts after watching that trailer.

 

[Hippasus]

Awesome thread. I am a mathematics major and had basically this same topic approved just last week for a final paper and presentation. The class is Writing in Mathematics. I am questioning whether irrational numbers (like sqrt 2) and transcendental numbers (like e) actually exist. They have infinitely complex decimal expansions. I think they do exist. My main premise at this point is that since infinite decimal expansions can be computed to an indefinite degree of accuracy and are not arbitrary with respect to individual digits, that this is evidence that such numbers are real. I think they can be considered abstract objects and that there is some mathematical substance undergirding the human apprehension of such matters through reason. I think I'll be arguing against a Kantian / later Wittgensteinian point of view whereby mathematics is merely a mode of apprehension for human cognition. I'm not sure if the latter two thinkers are saying mathematics is created, but at least something like that we can't know if such abstract objects exist since they are necessary to our thinking and we cannot see outside of this mode of apprehension. Thus, I think they would contend that we cannot judge whether they have an existence outside of our minds.

 

[x Tame Impala]

What does Physics say about infinity?

 

[JMCx4]

What does Physics say about infinity?

It has considered it ... and it wasn't happy ... Infinity Is a Beautiful Concept — And It's Ruining Physics

 

[x Tame Impala]

It has considered it ... and it wasn't happy ... Infinity Is a Beautiful Concept — And It's Ruining Physics

My understanding of infinity comes from college Calc 1-3 and Diff EQ, so hardly an expert and what I’m going to say could be wrong.

If I’m answering the thread question I’d say infinity only exists as a mathematical concept. An explanation for numbers extraordinarily big and small appearing in theory, but not in our physical world.

Mathematics says that when an arrow is shot there are an infinite number of of points in between where the arrow is shot and before the arrow hits its target. Physics says this isn’t true. Physics explains our reality where math can be sometimes limited to its theoretical concepts.

 

[beowulf]

My understanding of infinity comes from college Calc 1-3 and Diff EQ, so hardly an expert and what I’m going to say could be wrong.

If I’m answering the thread question I’d say infinity only exists as a mathematical concept. An explanation for numbers extraordinarily big and small appearing in theory, but not in our physical world.

Mathematics says that when an arrow is shot there are an infinite number of of points in between where the arrow is shot and before the arrow hits its target. Physics says this isn’t true. Physics explains our reality where math can be sometimes limited to its theoretical concepts.

But what about at a much larger scale? We have a known universe, what out technology allows us to see and this has grown by leaps and bounds over the last century as we created better telescopes and now satellites and other space based tech items that allow us to see farther around us than ever before. So far we don't seem to have found an end to this vast space that surrounds our planet. Will we ever find that the universe does have an end, a wall of some sort?

What about time does it go off into infinity or will the universe as we know someday end and all the stars go cold and finally it will collapse back into itself this ended both the infinite universe and the infinite time.

Physics also looks at dimensions and whether those are infinite. In the three dimensions we live in, some cosmologists believe the universe is not infinite and that it curves back onto itself in a manner that does make it finite but unbounded.

 

[x Tame Impala]

But what about at a much larger scale? We have a known universe, what out technology allows us to see and this has grown by leaps and bounds over the last century as we created better telescopes and now satellites and other space based tech items that allow us to see farther around us than ever before. So far we don't seem to have found an end to this vast space that surrounds our planet. Will we ever find that the universe does have an end, a wall of some sort?

What about time does it go off into infinity or will the universe as we know someday end and all the stars go cold and finally it will collapse back into itself this ended both the infinite universe and the infinite time.

Physics also looks at dimensions and whether those are infinite. In the three dimensions we live in, some cosmologists believe the universe is not infinite and that it curves back onto itself in a manner that does make it finite but unbounded.

View attachment 586411

1) Time is a concept so it can be infinitely represented

2) “Finite but unbounded” seems like the best answer to this question in general.

 

[x Tame Impala]

 

[beowulf]

It is indeed an interesting subject that we, as a species, may never know the answer to.

 

[adsfan]

Put this one here since it is a science question but also one of philosophy. Looks like an interesting documentary and I must admit have spent time in my life contemplating this idea of infinity. Does the universe really go on forever. Is the only real thing that is infinite is being dead? Do numbers go on forever?

Infinity is here in the Milwaukee area.

If you look at a map of Wauwatosa, 44th and 45th Streets meet.

I was always told that parallel lines meet at infinity. It must be there on the west side of the metro area.


My brother is a physicist by degree. I will have to consult with him, especially since he worked in astronomy.

 

[JMCx4]

View attachment 586411

The two middle lines of polygon projections reminded me of the Denys Fisher Toys Spirograph art kit that I had as a youngster many moons ago. Happier times.

 

[tarheelhockey]

I don't have much to add to the physics arguments. But I do suspect that the answers are closely tied to our human limitations in terms of how we can mentally conceive of things like directions, distances, numbers, etc.

If the edge of the universe acts anywhere near as weird as the things we are finding at the quantum level, the final answer might be "yes, but no, and it won't make sense to you anyway".

 

[Fourier]

Awesome thread. I am a mathematics major and had basically this same topic approved just last week for a final paper and presentation. The class is Writing in Mathematics. I am questioning whether irrational numbers (like sqrt 2) and transcendental numbers (like e) actually exist. They have infinitely complex decimal expansions. I think they do exist. My main premise at this point is that since infinite decimal expansions can be computed to an indefinite degree of accuracy and are not arbitrary with respect to individual digits, that this is evidence that such numbers are real. I think they can be considered abstract objects and that there is some mathematical substance undergirding the human apprehension of such matters through reason. I think I'll be arguing against a Kantian / later Wittgensteinian point of view whereby mathematics is merely a mode of apprehension for human cognition. I'm not sure if the latter two thinkers are saying mathematics is created, but at least something like that we can't know if such abstract objects exist since they are necessary to our thinking and we cannot see outside of this mode of apprehension. Thus, I think they would contend that we cannot judge whether they have an existence outside of our minds.

I am late to this thread but if one claims that sqrt{2} does not exist would that not mean that a right triangle with two sides of length one could not exist??

 

[Hippasus]

I am late to this thread but if one claims that sqrt{2} does not exist would that not mean that a right triangle with two sides of length one could not exist??

That sounds a lot like Platonic realism (ideal forms), and that is what I'm trying to promulgate. However, I think the algebraic means of describing the form is more rigorous than the geometric one. I'm trying to speak as much as possible to mathematics and less to philosophy and-or a general audience.

I think the issue about geometry vs. algebra and-or arithmetic in terms of being the most fundamental subdiscipline of mathematics is an interesting one. The ability of set theory to translate all of mathematics I think is evidence for arithmetic and-or algebra being more fundamental than geometry. Geometry perhaps relies more on something like intuition than classical and modern logic does.

 

[Fourier]

That sounds a lot like Platonic realism (ideal forms), and that is what I'm trying to promulgate. However, I think the algebraic means of describing the form is more rigorous than the geometric one. I'm trying to speak as much as possible to mathematics and less to philosophy and-or a general audience.

I think the issue about geometry vs. algebra and-or arithmetic in terms of being the most fundamental subdiscipline of mathematics is an interesting one. The ability of set theory to translate all of mathematics I think is evidence for arithmetic and-or algebra being more fundamental than geometry. Geometry perhaps relies more on something like intuition than classical and modern logic does.

I am not sure I understand the premise of your argument. The existence of sqrt{2} is perfectly reasonable under all standard models of set theory. I don't see an argument for why it might not exist but something like

1/[3*10^{10^1000000000000000000000000000000000000000}]

does.

Historically, a more troubling number was 0.

 

[Hippasus]

I am not sure I understand the premise of your argument. The existence of sqrt{2} is perfectly reasonable under all standard models of set theory. I don't see an argument for why it might not exist but something like

1/[3*10^{10^1000000000000000000000000000000000000000}]

does.

Historically, a more troubling number was 0.

I could try to refine the counterargument in the near future, but regarding the premise of sqrt(2), it is just an example and happens to be my favorite number. I picked it because there is a famous proof by contradiction concerning its existence. The scope of my paper / presentation is the existence of irrational numbers. I am also using Cantor's diagonal proof concerning the uncountability of the real numbers as well as the Riemann sum describing a means of computing the digits of e. I want to use some proof during the paper / presentation, but the e example is perhaps my most important one at this point, even though I don't have a proof concerning it. (This project is still ongoing.)

Zero perhaps is necessary for closure of arithmetical or algebraic systems--to satisfy certain equations, much like sqrt(2). But you're right, that would perhaps be a more contentious and unwieldy example.

Really, it is a philosophical topic, but one that is about mathematics. I am trying to use proof and logic more to try to get into the mathematics at the same time.

 

[JMCx4]

... Really, it is a philosophical topic, but one that is about mathematics. ...

 

[Fourier]

I could try to refine the counterargument in the near future, but regarding the premise of sqrt(2), it is just an example and happens to be my favorite number. I picked it because there is a famous proof by contradiction concerning its existence. The scope of my paper / presentation is the existence of irrational numbers. I am also using Cantor's diagonal proof concerning the uncountability of the real numbers as well as the Riemann sum describing a means of computing the digits of e. I want to use some proof during the paper / presentation, but the e example is perhaps my most important one at this point, even though I don't have a proof concerning it. (This project is still ongoing.)

Zero perhaps is necessary for closure of arithmetical or algebraic systems--to satisfy certain equations, much like sqrt(2). But you're right, that would perhaps be a more contentious and unwieldy example.

Really, it is a philosophical topic, but one that is about mathematics. I am trying to use proof and logic more to try to get into the mathematics at the same time.

I am curious what you mean by the bolded comment above. Are you referring to the existence of sqrt{2} or the fact that it is irrational.

As you probably know there are several ways to develop the real numbers from the rationals. They all have their pluses and minuses. The easiest one to get to sqrt{2} is to use Dedekind cuts. In this case, sqrt{2} corresponds to the cut...

A={r in Q | r<0 or r^2<2} B={r in Q| r>0, r^2>2}.

More to the point, Dedekind's method is a bear when it comes to the algebraic structure of the real line but it almost immediately provides you with the Least Upper Bound Property which gives you all of the core completeness properties in the most economical way.

Once you have completeness, the existence of sqrt{2} follows in many ways. As does the existence of e which you can simply define as the limit of the sequence,

S_n = 1^0/0!+ 1^1/1! + 1^2/2! + ... + 1^n/n!.

This sequence is clearly increasing, and it is easy to show that it is bounded above. Hence by the LUB its limit exists. You can also use this to show that e is irrational, though the proof that it is transcendental is much more difficult.

Zero has a very colorful history.

By the way, if you don't mind me asking where are you going to school? I was curious that you mentioned a "Writing in Mathematics" course. This is unusual but quite novel.

 

[Hippasus]

I am curious what you mean by the bolded comment above. Are you referring to the existence of sqrt{2} or the fact that it is irrational.

As you probably know there are several ways to develop the real numbers from the rationals. They all have their pluses and minuses. The easiest one to get to sqrt{2} is to use Dedekind cuts. In this case, sqrt{2} corresponds to the cut...

A={r in Q | r<0 or r^2<2} B={r in Q| r>0, r^2>2}.

More to the point, Dedekind's method is a bear when it comes to the algebraic structure of the real line but it almost immediately provides you with the Least Upper Bound Property which gives you all of the core completeness properties in the most economical way.

Once you have completeness, the existence of sqrt{2} follows in many ways. As does the existence of e which you can simply define as the limit of the sequence,

S_n = 1^0/0!+ 1^1/1! + 1^2/2! + ... + 1^n/n!.

This sequence is clearly increasing, and it is easy to show that it is bounded above. Hence by the LUB its limit exists. You can also use this to show that e is irrational, though the proof that it is transcendental is much more difficult.

Zero has a very colorful history.

By the way, if you don't mind me asking where are you going to school? I was curious that you mentioned a "Writing in Mathematics" course. This is unusual but quite novel.

Thank you very much! I might be able to use some of what you said for a better paper after I look into it more.

I am referring to Hippasus of Metapontum's proof showing sqrt(2) to be irrational. If we suppose a number corresponds to the hypotenuse of the triangle you mentioned, I take it to be an existence proof.

I am going to the University of Illinois at Chicago. I love it. That writing class is by far the easier of my two classes, it seems.

 

[Fourier]

Thank you very much! I might be able to use some of what you said for a better paper after I look into it more.

I am referring to Hippasus of Metapontum's proof showing sqrt(2) to be irrational. If we suppose a number corresponds to the hypotenuse of the triangle you mentioned, I take it to be an existence proof.

I am going to the University of Illinois at Chicago. I love it. That writing class is by far the easier of my two classes, it seems.

The irrationality of sqrt{2} is a very nice exercise that you can give to a variety of audience. I give the proof that e is irrational as an exercise to my first year Calculus class. It has a similar flavor, but of course with a much more sophisticated argument to obtain the contradiction.

If you want to take things a step further you can look at Robinson's non-standard real numbers and his approach to analysis.. The beauty of this is that it allows you to prove some of the heuristic results from Calculus rigourously using arithmetic arguments. In essennce it is the rigourous foundation for Leibnez's approach to Calculus.

 

[PromisedLand]

Put this one here since it is a science question but also one of philosophy. Looks like an interesting documentary and I must admit have spent time in my life contemplating this idea of infinity. Does the universe really go on forever. Is the only real thing that is infinite is being dead? Do numbers go on forever?

If you think of it...

Numbers

• infinity can be 1 to infinity (0,1,2,3,4, 5, .......)
• it can also be defined as how many time can something be divided in equal parts between 1 and 2? (answer: infinity: ex: 1/2, 1/4, 1/8, 1/16, etc...)
• how many decimals exist between 0 and 1? (infinity: ex: 0.1; 0.11, 0.12, 0.13, OR 0.01, 0.02, 0.03, OR 0.2, 0.21, 0.23, 0.24, etc....)
• How many decimals exist for number pi?
• etc...

Space itself is sometimes talked in terms of being infinite

Heck there is a hypothetical scenario that there might exist infinite number of universes (or multiverses).

all that said IMO infinity is less weird than quantum mechanics (at least the way we understand it now).