# Does nature abhor infinity?

#### Math Is Hard

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I can sort of deal with it conceptually, and even work with it mathematically - but are there any examples of it as a property of something in the physical universe? (I'm probably missing something obvious..)

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#### kant

Math Is Hard said:
I can sort of deal with it conceptually, and even work with it mathematically - but are there any examples of it as a property of something in the physical universe? (I'm probably missing something obvious..)
Hmm.... To me, your question is like asking if you can cut a piece of "stuff" down to infinite many parts. Is it possible? i don t know.

#### Math Is Hard

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kant said:
Hmm.... To me, your question is like asking if you can cut a piece of "stuff" down to infinite many parts. Is it possible? i don t know.
Thanks for your response. Now to the infinite cutting, I would say no. To cut a piece into infinitely small parts involves at some step cutting apart atoms, and as a member of PETA (People for the Ethical Treatment of Atoms) I am against this.

What I was thinking of is this.. is the universe infinitely big or just really, really big? Are there things in nature that are infinitely small? This I don't know. Can something be infinitely complex - no probably not, because the universe will expire before it evolves to infinite complexity.

But I am asking a stupid question here. I am not sure why I brought it up. I probably should have put it in the stupid quetions thread. :tongue:

#### Pengwuino

Gold Member
Wait a second, we CAN'T open up an atom????

#### Tychic

I suppose that I'm also missing something. But I think your question is very good (like many other simple questions).
Nature is finite insofar it is concrete. Insofar the connection of some properties involves mathematical derivations (particularly infinitesimal calculus) these relations are gained under the wrong assumption of infinity in nature. These relational properties have finite but irrational values. E.g. the radius of circle is irrational (and implicit infinite) when the diameter is given.
So the understanding of the concrete presupposes some understanding of the infinite (which is not concrete). Nature does not abhor infinity; it likes infinity as much as it likes to be understood.

#### Math Is Hard

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Thanks, Tychic. I thought that was a great answer. It really made me think - it's still making me think.

#### saltydog

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Math Is Hard said:
Thanks for your response. Now to the infinite cutting, I would say no. To cut a piece into infinitely small parts involves at some step cutting apart atoms, and as a member of PETA (People for the Ethical Treatment of Atoms) I am against this.

What I was thinking of is this.. is the universe infinitely big or just really, really big? Are there things in nature that are infinitely small? This I don't know. Can something be infinitely complex - no probably not, because the universe will expire before it evolves to infinite complexity.
Infinity is a metaphor for a singularity. Passing through such a singularity involves a qualitative change. The Universe is infinitely big, complex, nested, small only to the extent that such regressions involve qualitative changes in concepts: we keep looking for bigger and bigger and smaller and smaller, more and more complex until we reach singularities which change the qualitative nature of our conceptions rendering them no longer applicable. For example, we can consider swimming in cold water until we reach the freezing point. When it turns to ice, swimming is no longer applicable. I supsect this is the case with many concepts we have difficulty grasping such as infinitely big, small, and complex.

#### Gokul43201

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Gold Member
Nature does abhor singularities in that, at the very least, it makes them extremely rare.

Hawking once predicted that an observable naked (something not hidden behind an event horizon) singularity can not exist, and once again, was made to eat his words. But while they are theoretically possible, a naked singularity has not been observed (at least, I don't think so). But that may just be because singularities are shy ! "Clothed" singularities (or blackholes) of course, have been detected in the dozens (I think).

Also, infinities arising in physical theories are not happy things. Physicists look for theories that do not possess such infinities. In fact, if a "nice" theory has a few "not-nice" infinities (like the self-energy in early QED) in it, physicists will likely look for ways to kill the infinities rather than the theory. Look into renormalization.

#### nameta9

Infinity does exist. Physics and devices may some day be able to go down to the smallest dimension of 10^-40 mm (planck level). But 10^-10,000 mm exists as does any infinitely small dimension. We will never be able to see it or use it or interact with it, but this doesn't mean it is not there. The infinitely small is there, we will never have any idea what happens at those dimensions. 10^-infinity mm exists but not for us!

It is like the earth's mantle. Did you know that from 500 km under the earth to 3000 km there is matter and rocks ? Will we ever be able to explore it completely ? Will we ever see all the trillions of details of all the rock structures, chemical compostions in the earth's mantle ? Many scientists are concentrated on strings and abstractions and technology, but we still haven't even observed and we still do not even know 90 % of our own earth!

#### Chronos

Gold Member
No confirmed observations of an 'infinite' amount of anything has yet been observed, to my knowledge. My guess, it would probably take an infinite amount of time to confirm any such suspected observation. Mathematical infinites, of course, lurk behind every corner. And they are maddening. We have numerous examples of finite numbers that represent the sum of an infinite number of non-zero quantities - like pi [be warned, I'm just dying for someone to ask to see that formula]. How can that be explained logically?

#### Tychic

saltydog said:
Infinity is a metaphor for a singularity. Passing through such a singularity involves a qualitative change.
In what sense a singularity is infinite? Is it infinite because it is incountable? I do not think so. Is it infinite because there is an asymptotic function to some value in space but the function itself does not reach this point (e.g. 1/x and the y-axis)? Do you mean something like this by a qualitative change?
The Universe is infinitely big, complex, nested, small only to the extent that such regressions involve qualitative changes in concepts: we keep looking for bigger and bigger and smaller and smaller, more and more complex until we reach singularities which change the qualitative nature of our conceptions rendering them no longer applicable. For example, we can consider swimming in cold water until we reach the freezing point. When it turns to ice, swimming is no longer applicable.
Why do you call this an infinity? It is a border for the application of a concept, but is it also an infinity?
nameta9 said:
Infinity does exist. Physics and devices may some day be able to go down to the smallest dimension of 10^-40 mm (planck level). But 10^-10,000 mm exists as does any infinitely small dimension.
I’m not so sure about that. Quantum physics has its name from the quantum that is the discrete smallest entity. In mathematics you always can detect a smaller number but the question of this thread is whether and in what degree this mathematical feature (and other mathematical procedures that lead to infinity) can be applied in physics. I think in concrete nature there is always a border. Hence there is no concrete infinity, nothing concrete is infinite big, infinite small, or moves with infinite velocity (although there may be a velocity higher then the speed of light c)
I agree with Chronos (and perhaps also with Math Is Hard) concerning the last post:
Chronos said:
No confirmed observations of an 'infinite' amount of anything has yet been observed, to my knowledge. My guess, it would probably take an infinite amount of time to confirm any such suspected observation. Mathematical infinites, of course, lurk behind every corner. And they are maddening. We have numerous examples of finite numbers that represent the sum of an infinite number of non-zero quantities - like pi [be warned, I'm just dying for someone to ask to see that formula]. How can that be explained logically?
One small divergence: I’m not sure whether there can be a logical explanation for this. If you want to explore the connection of mathematical infinity and concreteness by logic you take another infinite discipline for exploration. But the main part of the first question is: How can an infinite system explore?
I’m astonished about this feature since many years. Do you have an idea of an non logical explanation of this feature?

#### nameta9

The problem is the definition of "existence". 10^-infinity exists but we can't interact with it or see it, it would seem to not exist. But it does. Unless you want to change what exists mean. Existence is a tough metaphysical concept.

#### Tychic

nameta9, do you think that all entities and everything that you can address exist? That would be a metaphysical thesis that is controversial.
Second: Granted that infinity exists. Does infinity exist in nature and does it exist as something concrete? Are there some infinite things?

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#### nameta9

It all depends on what "exists" means. If it is something that is present but that we can never interact with, then whether it exists or not is indifferent. I think infinitely small exists, but we can't do anything with it. It is present and in that sense exists. We can't use it, or see it, or interact with 10^-455 mm. Many philosophers have tried to understand and pinpoint down "existence". It may never be possible.

As a corollary, since infinite (small) exists then an infinite-infinite univere can fit in it, hence every conceivable universe exists. see :

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#### saltydog

Homework Helper
Tychic said:
In what sense a singularity is infinite?
I would ask, "in what sense is infinity a singularity?" My notion of a singularity is the point which separates two qualitatively different states of a system. The extent of this difference offers a very wide spectrum: water changing to ice, a heart attack, a collapsing bridge, an avalanche, an explosion, apes changing to man, a supernova, and the Big Bang itself: The system bifurcates as it passes through the singularity to a qualitatively different state.

And what of an infinite sum? Does a convergent one ever really get to the value we assign it? That act, of "assigning" the sum to a distinct value, the limiting one, is the passage, I think, through the singularity we call "an infinitely large number of additions".

And of a non-convergent one? Just a different singularity I suppose. Perhaps one day we may be better able to grasp this sort.

I interpret infinitely large or small from this perspective by claiming "size" looses meaning at some point or rather SHOULD loose meaning in order to obtain a better understanding of the phenomenon although that seems difficult to comprehend for me. The same could be said about infinitely old and infinitely complex. At some point in the analysis a singularity may be reached rendering old concepts meaningless and requiring other qualitatively different ones for greater understanding.

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#### nameta9

Size can't loose meaning at any point. It just keeps on getting smaller forever. If it did loose meaning then what would it become, larger than the previous "size" ? A possible solution is to think of the universe as circular. At 10^-1000 mm we have the universe all over again and as you get smaller you can see the earth again and then ourselves and then atoms and then the cycle repeats forever. So we would in effect be made up of ourselves in this infinite loop. It would be like an infinite recursion where A is composed of B and B is composed of A etc.

Of course in between you could have all kinds of structures maybe at 10^-10000000000 mm you could have all kinds of oddball universe and matter, but the bottom line is that elementary particles would not really exist anymore and even size as a concept would no longer make much sense since all is made up of itself forever.

#### saltydog

Homework Helper
nameta9 said:
Size can't loose meaning at any point. It just keeps on getting smaller forever. If it did loose meaning then what would it become, larger than the previous "size" ? A possible solution is to think of the universe as circular. At 10^-1000 mm we have the universe all over again and as you get smaller you can see the earth again and then ourselves and then atoms and then the cycle repeats forever. So we would in effect be made up of ourselves in this infinite loop. It would be like an infinite recursion where A is composed of B and B is composed of A etc.

Of course in between you could have all kinds of structures maybe at 10^-10000000000 mm you could have all kinds of oddball universe and matter, but the bottom line is that elementary particles would not really exist anymore and even size as a concept would no longer make much sense since all is made up of itself forever.
Yes Nameta, interesting. You know, what we need is a singularity in understanding: You know how discovery works: A new idea emerges and then is elaborated upon and refined. Later, another one is defined and the cycle repeats. These "new" ideas, revolutions like Relativity, QM, Evolution, Calculus, integrated circuits, the digital computer and so on represent bifurcations in our understanding of Nature: Someone comes up with a "singular idea" and thus begins a revolution.

I wonder what singular ideas will take us out of the present quagmire Cosmology now is experiencing? I look outside of my window at the world and wait for a modern-day Kepler.

#### nameta9

I read some time back that fractals do something similar. The details go on forever and size as we know it doesn't even make any sense since the details sometimes repeat all over again as magnification continues. It is as if size is only a local perception we have but is not a global property of matter. Very strange indeed. I also think that string theories have systems that contain themselves (inversions ?) and are contained within themselves at the same time.

In my recursive universe A is composed of B, and B is composed of A but it may also be that the loop ends there and the last A is equal to the first.

#### Tychic

Hi saltydog and nameta9 and all others,
nameta9 said:
In my recursive universe A is composed of B, and B is composed of A but it may also be that the loop ends there and the last A is equal to the first.
This reciprocal definition could also be implemented in some other places of the universe that do not have any connection to infinity (e.g. between receptive and effective properties in Gregg Rosenberg's causal theory) and I do not think that this will help us in talking about the infinity in nature.

I think we have three concepts of infinity in nature for discussion:
1) SALTYDOG: a singularity, a point where a concept loose its meaning, e.g. there is infinite velocity in a black hole because to speak of velocity makes no sense inside the black hole
2) NAMETA9: infinite is that there always is something bigger, smaller, faster as all discrete values and borders of the concrete. (It is not important whether this infinite is verified in some experiment or takes part in some causal chains.)
3) TYCHIC: The infinite is something mathematical and exists in nature insofar as the nature is understood, but it does not exist in nature as something concrete.

Some tests:
A) Difference between countable infinity and uncountable infinity:
- accepted when considering infinity 3)
- infinity 2) there is only countable infinity,
- infinity 1) I would suppose is talked about uncountable infinity​
B) Does this infinity exist in nature in the sense that it makes a causal difference?
- infinity 1) yes, the big bang e.g. was causally very important
- infinity 2) this infinity makes no causal difference. Would you agree, nameta9 that in this sense infinity does not exist in nature?
- infinity 3) it seems that this infinity also makes no causal difference because it is only for understanding. But one could argue that if there would not be any such infinity the causal connections would not exist​
C) Are these infinities rare?
- Infinity 1) SALTYDOG would deny this but Gokul43201 would accept that there are only few singularities (Why do you think nature tries to hide its infinities and “clothes” the “shy” black holes?)
- infinities 2 and 3 are not rare.​
D) Connection between the infinite and the rest of the nature
- Infinity 2) has no special connection to the empirical data.
- Infinity 1) and 3) are central feaures for explanation the connection is in case 1 an empirical, but in case 3 a very difficult question that cannot be resolved by empirical and logical methods alone. I’m very interested in this connection.
Have I misunderstood or overseen something? Can someone help me in the last question?

#### nameta9

You are correct, this infinity makes no causal difference (I don't know what existence means...). Another possible model is that A is made up of 2 A and each A is made up of 2 A forever.

A

after magnification is composed of (or has elementary particles the same but smaller (?) ) so you would see A under the microscope as

A A

then again after magnification each A is composed of (or has elementary particles the same but smaller (?) ) so you would see A A under the microscope as

A A A A

and so on forever. Maybe in 3 dimensions as a lattice. The math could be like :

A<>A # A is not equal to A (using BASIC language as not equal and perl
comment symbol # ;)

A>A

A<A

A ) A # A contains A

A ( A # A is contained in A

all these statements valid at the same time.

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#### saltydog

Homework Helper
Hey guys. Perhaps I'm abusing the term "singularity". I don't know for sure. My idea is any abrupt qualitative change passes through a singularity: If I gradually move a vase across the top of a desk, nothing much happens until I get very close to some critical point about half-way over the edge. Just a very slight nudge and it falls to the floor and breaks. That critical point between existing on the desk and falling to the floor is a singularity as I see it. Is this an incorrect usage of the term?

Applying that concept to an infinite sum or other infinities in Mathematics is a bit awkward for me: On one side of the singularity we have an infinite sum. On the other side we have a single value it converges to. Not sure how to defend the argument: How is that different from just a finite sum on one side and the value of that sum on the other?

I know a black hole is the common example for a singularity. I imagine the physical Universe we observe on one side of that singularity and perhaps a qualitatively different Universe on the other side. This is also how I envision the Big Bang: A singularity separating our existence from another pre-existence complete with its own qualitatively different physics.
It's for this reason I suspect one day we may find an intimate connection between black holes and the origin of our Universe. Just a hunch.

Thus I invoke this usage of singularity, as a qualitatively different change, to resolve in my mind some of the paradoxes we see in our world: As we invent imaginary numbers to resolve certain algebraic difficulties, I call upon singularities to resolve paradoxes.

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Staff Emeritus
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Dearly Missed
saltydog, what you are calling a singularity is actually called a critical point. For exmple 200 degrees Fahrenheit is a critical point for water (at standard conditions), where it changes from liquid to gas form. The study of critical points and the attempt to calculate them from firstr principles is a hot area these days.

The same idea has been popularized in social systems as a "tipping point".

A singularity is a state in which some value increases without bound; "goes to infinity" in informal speech. The function $$\frac{1}{(x-5)}$$ has a singularity at x=5.

#### saltydog

Homework Helper
saltydog, what you are calling a singularity is actually called a critical point. For exmple 200 degrees Fahrenheit is a critical point for water (at standard conditions), where it changes from liquid to gas form. The study of critical points and the attempt to calculate them from firstr principles is a hot area these days.

The same idea has been popularized in social systems as a "tipping point".

A singularity is a state in which some value increases without bound; "goes to infinity" in informal speech. The function $$\frac{1}{(x-5)}$$ has a singularity at x=5.
Ok, very good. I'll adjust my nomenclature.

#### nameless

This (infinite-infinite universe hypothesis) is not what you'll be reading in the Science Journals in 10-15 years.
Anyone wanna bet?

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