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infinity must die

slo1
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3/2/2015 6:47:22 PM
Posted: 1 year ago
http://blogs.discovermagazine.com...
I was seduced by infinity at an early age. Georg Cantor"s diagonality proof that some infinities are bigger than others mesmerized me, and his infinite hierarchy of infinities blew my mind. The assumption that something truly infinite exists in nature underlies every physics course I"ve ever taught at MIT"and, indeed, all of modern physics. But it"s an untested assumption, which begs the question: Is it actually true?
slo1
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3/2/2015 6:48:42 PM
Posted: 1 year ago
I heard this guy on npr promoting some concepts that he believes is hindering science. A very interesting thought to kill infinity.
RuvDraba
Posts: 6,033
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3/2/2015 8:04:28 PM
Posted: 1 year ago
At 3/2/2015 6:47:22 PM, slo1 quoted:
The assumption that something truly infinite exists in nature underlies every physics course I"ve ever taught at MIT"and, indeed, all of modern physics. But it"s an untested assumption, which begs the question: Is it actually true?

Slo, I used to work as a scientist and mathematician in formal systems and automated reasoning, so the question of provability in mathematical systems and their applicability to the real world was directly relevant to my work.

In that respect, I think 'true' isn't quite the right question to ask. Let me offer some background.

Mathematically, we know that the infinity representing (say) the real numbers (all the fractions, and square roots and pi and other numbers we use in normal space) is a different sort of infinity to the one representing all the finite sentences in the English language, say.

If you tried to count all the English sentences we might ever write, there would be an infinite number of them, and if you tried to count the real numbers, there are an infinite number of them too. But in counting English sentences, you can put them in (say) an alphabetical order so that you always know what the next sentence is in your list, and any sentence in the list will eventually be counted.

But in counting real numbers, most of them would not be counted, because there aren't just infinitely many; there are infinitely many between any two numbers, so however you pick the 'next' number, there are always infinitely many between that one and the last one, that you'll miss out.

So mathematicians say that English sentences are countably infinite, while real numbers are uncountably infinity.

Freaky, huh? :D

Yet real numbers offer a very powerful model for physics, because they let us cut space and time as finely as we want, they let us draw perfect circles and use all kinds of beautiful geometries to model it. And for experimental purposes, it seems to be reliable. For example, pure mathematics predicts that if you measure the ratio of a circle's circumference to its diameter, you'll get an infinitely long decimal number that never repeats, and can't be written as a fraction. And to the precision that we can measure such things, experiment bears that out.

But is it the best model for physics?

It might not be. Because if matter and energy can be quantised, then can space and time be quantised too? If they can, then might space and time work more like the picsels on a computer screen, so that circles are never perfectly smooth, and movement is always slightly jerky?

If so, we might be better off not using real numbers. Yet as far as I'm aware, we don't know that -- but I'm not clear that we know it's not either.

So it may be that real numbers are a good model for physics: powerful to use, and true enough to work with for most purposes, but not necessarily the most accurate. If so, we'll probably keep using them unless and until we find something better.

I hope that may help.
slo1
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3/3/2015 6:58:17 AM
Posted: 1 year ago
At 3/2/2015 8:04:28 PM, RuvDraba wrote:
At 3/2/2015 6:47:22 PM, slo1 quoted:
The assumption that something truly infinite exists in nature underlies every physics course I"ve ever taught at MIT"and, indeed, all of modern physics. But it"s an untested assumption, which begs the question: Is it actually true?

Slo, I used to work as a scientist and mathematician in formal systems and automated reasoning, so the question of provability in mathematical systems and their applicability to the real world was directly relevant to my work.

In that respect, I think 'true' isn't quite the right question to ask. Let me offer some background.

Mathematically, we know that the infinity representing (say) the real numbers (all the fractions, and square roots and pi and other numbers we use in normal space) is a different sort of infinity to the one representing all the finite sentences in the English language, say.

If you tried to count all the English sentences we might ever write, there would be an infinite number of them, and if you tried to count the real numbers, there are an infinite number of them too. But in counting English sentences, you can put them in (say) an alphabetical order so that you always know what the next sentence is in your list, and any sentence in the list will eventually be counted.

But in counting real numbers, most of them would not be counted, because there aren't just infinitely many; there are infinitely many between any two numbers, so however you pick the 'next' number, there are always infinitely many between that one and the last one, that you'll miss out.

So mathematicians say that English sentences are countably infinite, while real numbers are uncountably infinity.

Freaky, huh? :D

Yet real numbers offer a very powerful model for physics, because they let us cut space and time as finely as we want, they let us draw perfect circles and use all kinds of beautiful geometries to model it. And for experimental purposes, it seems to be reliable. For example, pure mathematics predicts that if you measure the ratio of a circle's circumference to its diameter, you'll get an infinitely long decimal number that never repeats, and can't be written as a fraction. And to the precision that we can measure such things, experiment bears that out.

But is it the best model for physics?

It might not be. Because if matter and energy can be quantised, then can space and time be quantised too? If they can, then might space and time work more like the picsels on a computer screen, so that circles are never perfectly smooth, and movement is always slightly jerky?

If so, we might be better off not using real numbers. Yet as far as I'm aware, we don't know that -- but I'm not clear that we know it's not either.

So it may be that real numbers are a good model for physics: powerful to use, and true enough to work with for most purposes, but not necessarily the most accurate. If so, we'll probably keep using them unless and until we find something better.

I hope that may help.

thanks for the thoughts. Just one clatification to readers though. Your example of an infinite numbers of english sentences is flawed. It is impossible to have an infinite set of combinations of english words that would have enought meaning to be classified as a sentence.

It mirrors the authors point in that when you extract the cocept of infinity to something in the real world it becomes nonsensical.

Also to the authors point we have no empirical example of any thing infinite in the physical world.

Take pi as your other example. If the radius of a circle is 7 and I calculate the circumference, the circumference is 43.98..... , an infinitely precise decimal. If there is a limit on how small things can be, pi becomes incorrect in its use to calculate physical properties such as circumference.

While I think it may be premature to throw infinity out in the physical sciences, I agree with the author that it should nit be such a readily accepted assumption.
UndeniableReality
Posts: 1,897
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3/3/2015 9:31:28 AM
Posted: 1 year ago
At 3/3/2015 6:58:17 AM, slo1 wrote:
At 3/2/2015 8:04:28 PM, RuvDraba wrote:
At 3/2/2015 6:47:22 PM, slo1 quoted:
The assumption that something truly infinite exists in nature underlies every physics course I"ve ever taught at MIT"and, indeed, all of modern physics. But it"s an untested assumption, which begs the question: Is it actually true?

Slo, I used to work as a scientist and mathematician in formal systems and automated reasoning, so the question of provability in mathematical systems and their applicability to the real world was directly relevant to my work.

In that respect, I think 'true' isn't quite the right question to ask. Let me offer some background.

Mathematically, we know that the infinity representing (say) the real numbers (all the fractions, and square roots and pi and other numbers we use in normal space) is a different sort of infinity to the one representing all the finite sentences in the English language, say.

If you tried to count all the English sentences we might ever write, there would be an infinite number of them, and if you tried to count the real numbers, there are an infinite number of them too. But in counting English sentences, you can put them in (say) an alphabetical order so that you always know what the next sentence is in your list, and any sentence in the list will eventually be counted.

But in counting real numbers, most of them would not be counted, because there aren't just infinitely many; there are infinitely many between any two numbers, so however you pick the 'next' number, there are always infinitely many between that one and the last one, that you'll miss out.

So mathematicians say that English sentences are countably infinite, while real numbers are uncountably infinity.

Freaky, huh? :D

Yet real numbers offer a very powerful model for physics, because they let us cut space and time as finely as we want, they let us draw perfect circles and use all kinds of beautiful geometries to model it. And for experimental purposes, it seems to be reliable. For example, pure mathematics predicts that if you measure the ratio of a circle's circumference to its diameter, you'll get an infinitely long decimal number that never repeats, and can't be written as a fraction. And to the precision that we can measure such things, experiment bears that out.

But is it the best model for physics?

It might not be. Because if matter and energy can be quantised, then can space and time be quantised too? If they can, then might space and time work more like the picsels on a computer screen, so that circles are never perfectly smooth, and movement is always slightly jerky?

If so, we might be better off not using real numbers. Yet as far as I'm aware, we don't know that -- but I'm not clear that we know it's not either.

So it may be that real numbers are a good model for physics: powerful to use, and true enough to work with for most purposes, but not necessarily the most accurate. If so, we'll probably keep using them unless and until we find something better.

I hope that may help.

thanks for the thoughts. Just one clatification to readers though. Your example of an infinite numbers of english sentences is flawed. It is impossible to have an infinite set of combinations of english words that would have enought meaning to be classified as a sentence.

It mirrors the authors point in that when you extract the cocept of infinity to something in the real world it becomes nonsensical.

Also to the authors point we have no empirical example of any thing infinite in the physical world.

Take pi as your other example. If the radius of a circle is 7 and I calculate the circumference, the circumference is 43.98..... , an infinitely precise decimal. If there is a limit on how small things can be, pi becomes incorrect in its use to calculate physical properties such as circumference.

While I think it may be premature to throw infinity out in the physical sciences, I agree with the author that it should nit be such a readily accepted assumption.

I'm not sure there's really an assumption in the scientific community that there are real infinities in existence. Infinity it just useful in many calculations and for mathematical modelling with arbitrary precision (as you are probably already aware).
RuvDraba
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3/3/2015 11:15:12 AM
Posted: 1 year ago
At 3/3/2015 6:58:17 AM, slo1 wrote:
thanks for the thoughts. Just one clatification to readers though. Your example of an infinite numbers of english sentences is flawed. It is impossible to have an infinite set of combinations of english words that would have enought meaning to be classified as a sentence.

That's not quite true, Slo. We can do it with a simple subset of English. Consider:

The cat sat on the mat.
The cat and the cat sat on the mat.
The cat and the cat and the cat sat on the mat.
...
(The cat and) ^ n-1 the cat sat on the mat
...

This is a simple sequence of English sentences, all of which are meaningful and finite length, yet the sequence grows as long as you want, and never terminates.

There are other ways of producing such sequences, but if we can do it with with a subset of English, then obviously we can do it with the whole of the language too.

to the authors point we have no empirical example of any thing infinite in the physical world.

The author is right. Empirical testing has to work with finite steps in finite time. So even if some sequence (like my cats above) has the potential to become infinite, we can't confirm empirically that it ever does. Likewise, if space is infinitely divisible, we can't prove that. It's model-theoretic only.

Does that hurt our theories though?

It tends not to, because rather than picking on infinity as an object or property, physics tends to work around it. Here's an example.

Einstein's theory of relativistic mass says an object gets heavier and heavier as it approaches the speed of light. The equation for this is:
Mass at speed = Mass at rest / square root (1 - speed squared / lightspeed squared)

So mass at velocity equals resting mass divided by the square root of one minus speed as a fraction of light-speed squared. As speed approaches lightspeed, that fraction becomes close to one, and so we get closer to dividing the resting mass by zero, which gives us infinity.

However, physicists never actually divide by zero, because they never let velocity equal lightspeed, so they never get infinity. Inifinity remains a philosophical concept, rather than a physical object to work with.

At 3/3/2015 6:58:17 AM, slo1 wrote:
Take pi as your other example. If the radius of a circle is 7 and I calculate the circumference, the circumference is 43.98..... , an infinitely precise decimal. If there is a limit on how small things can be, pi becomes incorrect in its use to calculate physical properties such as circumference.

Or rather, theoretical pi predicts the value of actual pi, as long as we stay within certain limits.

While I think it may be premature to throw infinity out in the physical sciences, I agree with the author that it should nit be such a readily accepted assumption.

As I hope to have shown, it isn't an assumption. You can do virtually everything in physics with arbitrarily large, finite models that don't use infinity. However, as I mentioned earlier, if it turns out that space and time are quantised, there may be 'picsel' effects down at the level of the quanta which would make us need to restrict our model when working at that scale. That would affect pi, but many other calculations too. Yet today, physical calculations don't use pi as an infinite expansion anyway. They always calculate it to some limit of precision, so we're already acting as though infinity doesn't exist.

The worst place where reasoning about infinity is done complacently is actually philosophy. In particular, god-proofs and origin-proofs use reasoning about infinity all the time. So if we don't think infinity should be tossed about so casually, then there may be some questions we can't ask about -- like What Happened Before the Big Bang, or What is the Greatest Thing Man Could Ever Conceive.

I hope that may be useful.
NoMagic
Posts: 507
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3/9/2015 7:29:23 PM
Posted: 1 year ago
At 3/3/2015 9:31:28 AM, UndeniableReality wrote:
At 3/3/2015 6:58:17 AM, slo1 wrote:
At 3/2/2015 8:04:28 PM, RuvDraba wrote:
At 3/2/2015 6:47:22 PM, slo1 quoted:
The assumption that something truly infinite exists in nature underlies every physics course I"ve ever taught at MIT"and, indeed, all of modern physics. But it"s an untested assumption, which begs the question: Is it actually true?

Slo, I used to work as a scientist and mathematician in formal systems and automated reasoning, so the question of provability in mathematical systems and their applicability to the real world was directly relevant to my work.

In that respect, I think 'true' isn't quite the right question to ask. Let me offer some background.

Mathematically, we know that the infinity representing (say) the real numbers (all the fractions, and square roots and pi and other numbers we use in normal space) is a different sort of infinity to the one representing all the finite sentences in the English language, say.

If you tried to count all the English sentences we might ever write, there would be an infinite number of them, and if you tried to count the real numbers, there are an infinite number of them too. But in counting English sentences, you can put them in (say) an alphabetical order so that you always know what the next sentence is in your list, and any sentence in the list will eventually be counted.

But in counting real numbers, most of them would not be counted, because there aren't just infinitely many; there are infinitely many between any two numbers, so however you pick the 'next' number, there are always infinitely many between that one and the last one, that you'll miss out.

So mathematicians say that English sentences are countably infinite, while real numbers are uncountably infinity.

Freaky, huh? :D

Yet real numbers offer a very powerful model for physics, because they let us cut space and time as finely as we want, they let us draw perfect circles and use all kinds of beautiful geometries to model it. And for experimental purposes, it seems to be reliable. For example, pure mathematics predicts that if you measure the ratio of a circle's circumference to its diameter, you'll get an infinitely long decimal number that never repeats, and can't be written as a fraction. And to the precision that we can measure such things, experiment bears that out.

But is it the best model for physics?

It might not be. Because if matter and energy can be quantised, then can space and time be quantised too? If they can, then might space and time work more like the picsels on a computer screen, so that circles are never perfectly smooth, and movement is always slightly jerky?

If so, we might be better off not using real numbers. Yet as far as I'm aware, we don't know that -- but I'm not clear that we know it's not either.

So it may be that real numbers are a good model for physics: powerful to use, and true enough to work with for most purposes, but not necessarily the most accurate. If so, we'll probably keep using them unless and until we find something better.

I hope that may help.

thanks for the thoughts. Just one clatification to readers though. Your example of an infinite numbers of english sentences is flawed. It is impossible to have an infinite set of combinations of english words that would have enought meaning to be classified as a sentence.

It mirrors the authors point in that when you extract the cocept of infinity to something in the real world it becomes nonsensical.

Also to the authors point we have no empirical example of any thing infinite in the physical world.

Take pi as your other example. If the radius of a circle is 7 and I calculate the circumference, the circumference is 43.98..... , an infinitely precise decimal. If there is a limit on how small things can be, pi becomes incorrect in its use to calculate physical properties such as circumference.

While I think it may be premature to throw infinity out in the physical sciences, I agree with the author that it should nit be such a readily accepted assumption.

I'm not sure there's really an assumption in the scientific community that there are real infinities in existence. Infinity it just useful in many calculations and for mathematical modelling with arbitrary precision (as you are probably already aware).

Undeniable reality, you seem like a person worth asking this question to. I've heard on several occasions, someone (generally of academic variety) say infinities cannot exist in nature. Yet I've never heard as to why this is the case. Do you know why those who hold this view think that?
Based on my own thinking, I think space is probably infinite. I cannot figure out how it can stop. If I were to imagine space with an edge, I must ask the question what is beyond the edge, thinking the edge is a divider between two regions. This seems to lead to an infinite line of "well what is beyond that edge?" Any insights here?
UndeniableReality
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3/9/2015 9:50:41 PM
Posted: 1 year ago
At 3/9/2015 7:29:23 PM, NoMagic wrote:
At 3/3/2015 9:31:28 AM, UndeniableReality wrote:

I'm not sure there's really an assumption in the scientific community that there are real infinities in existence. Infinity it just useful in many calculations and for mathematical modelling with arbitrary precision (as you are probably already aware).

Undeniable reality, you seem like a person worth asking this question to. I've heard on several occasions, someone (generally of academic variety) say infinities cannot exist in nature. Yet I've never heard as to why this is the case. Do you know why those who hold this view think that?
Based on my own thinking, I think space is probably infinite. I cannot figure out how it can stop. If I were to imagine space with an edge, I must ask the question what is beyond the edge, thinking the edge is a divider between two regions. This seems to lead to an infinite line of "well what is beyond that edge?" Any insights here?

I can only speak for myself, but I don't know whether there are real infinities in nature, and I doubt that any person can make that determination right now. We can definitely imagine what I think are possible infinities, and I'm not sure there's a specific reason to think they can't exist. For example, I'm not aware of any definitive demonstration that there is a planck time (a fundamental unit of time that is indivisible, like planck length). If there is no planck time, then there are certain real infinities that must exist, such as the number of moments between any two moments.

I've definitely heard scientists, physicists included, talk about possible real infinities (though in hypothetical cases). To bring in your question, I don't think most physicists think that space is infinite (current cosmology suggests that space was finite in size a finite amount of time ago, and has been expanding at a finite rate the entire time, so that should result in a space time that is finite in space and finite in time) but I don't think time has been determine to be finite in the future. We could imagine that the universe is infinite in the future even if it becomes completely lifeless due to entropy. That would be a real infinite in nature. Or would the universe evaporate? Or be destroyed somehow? No idea.

More related to your example, we've all heard of the multiverse concept. Let's just talk about the entire family of concepts which posit that there is something beyond the universe. Then whether or not our universe is finite in all ways, there could be infinities in whatever is beyond. So you're right that there could be something beyond our universe which could be infinite, or something beyond that ad infinitum. I don't know if we would call the beyond 'space' though, and it at least wouldn't be our 'spacetime'. But that's just semantics. I hope I've interpreted your example correctly.

That being said, we haven't actually found a real infinity, and we've usually found that things we thought were infinite probably aren't. We used to think space was continuous (that there's no planck length) and that space was infinitely large, but most physicists don't think so any more (or so I hear). So there might be the expectation that all possible real infinities will be found finite in time. Of course, this alone doesn't justify the claim that there are no real infinities in nature.

So, I don't know why some people might say there are no real infinities in nature. They either knows something about this question that I don't, which is easily achievable, or they are over-generalizing. I think people over-generalize a lot and that human brains are susceptible to forming beliefs that are stronger than the evidence they have suggests.

Sorry for the wall of text and if some of the parenthetical statements were pedantic. I didn't want to assume we had entirely the same vocabulary.
NoMagic
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3/10/2015 9:13:46 PM
Posted: 1 year ago
At 3/9/2015 9:50:41 PM, UndeniableReality wrote:
At 3/9/2015 7:29:23 PM, NoMagic wrote:
At 3/3/2015 9:31:28 AM, UndeniableReality wrote:

I'm not sure there's really an assumption in the scientific community that there are real infinities in existence. Infinity it just useful in many calculations and for mathematical modelling with arbitrary precision (as you are probably already aware).

Undeniable reality, you seem like a person worth asking this question to. I've heard on several occasions, someone (generally of academic variety) say infinities cannot exist in nature. Yet I've never heard as to why this is the case. Do you know why those who hold this view think that?
Based on my own thinking, I think space is probably infinite. I cannot figure out how it can stop. If I were to imagine space with an edge, I must ask the question what is beyond the edge, thinking the edge is a divider between two regions. This seems to lead to an infinite line of "well what is beyond that edge?" Any insights here?

I can only speak for myself, but I don't know whether there are real infinities in nature, and I doubt that any person can make that determination right now. We can definitely imagine what I think are possible infinities, and I'm not sure there's a specific reason to think they can't exist. For example, I'm not aware of any definitive demonstration that there is a planck time (a fundamental unit of time that is indivisible, like planck length). If there is no planck time, then there are certain real infinities that must exist, such as the number of moments between any two moments.

I've definitely heard scientists, physicists included, talk about possible real infinities (though in hypothetical cases). To bring in your question, I don't think most physicists think that space is infinite (current cosmology suggests that space was finite in size a finite amount of time ago, and has been expanding at a finite rate the entire time, so that should result in a space time that is finite in space and finite in time) but I don't think time has been determine to be finite in the future. We could imagine that the universe is infinite in the future even if it becomes completely lifeless due to entropy. That would be a real infinite in nature. Or would the universe evaporate? Or be destroyed somehow? No idea.

More related to your example, we've all heard of the multiverse concept. Let's just talk about the entire family of concepts which posit that there is something beyond the universe. Then whether or not our universe is finite in all ways, there could be infinities in whatever is beyond. So you're right that there could be something beyond our universe which could be infinite, or something beyond that ad infinitum. I don't know if we would call the beyond 'space' though, and it at least wouldn't be our 'spacetime'. But that's just semantics. I hope I've interpreted your example correctly.

That being said, we haven't actually found a real infinity, and we've usually found that things we thought were infinite probably aren't. We used to think space was continuous (that there's no planck length) and that space was infinitely large, but most physicists don't think so any more (or so I hear). So there might be the expectation that all possible real infinities will be found finite in time. Of course, this alone doesn't justify the claim that there are no real infinities in nature.

So, I don't know why some people might say there are no real infinities in nature. They either knows something about this question that I don't, which is easily achievable, or they are over-generalizing. I think people over-generalize a lot and that human brains are susceptible to forming beliefs that are stronger than the evidence they have suggests.

Sorry for the wall of text and if some of the parenthetical statements were pedantic. I didn't want to assume we had entirely the same vocabulary.

Thanks for your response. I will add a few things though. I've only heard that no infinites exist in nature but have never heard the justification for this statement. Which seems to be that same thing you've heard. So I will continue to wonder what motivates that statement and to generally think it is wrong.
Also, I pay quite a bit of attention to cosmology and physics. I'm don't think the notion of the Big Bang being the beginning of "everything" is largely considered to be correct in both the physics or cosmology community. I think the general public thinks that. Perhaps for historical reasons, like the Big Bang has historically been presented that way. But, I've listened to many physicists speak much more carefully regarding that. Alan Guth, MIT, suspects the universe is eternal. Sean Carrol, Caltech, suspects the same and many others. The view the Big Bang represents the beginning is probably held very tentatively in the physics community if not just set aside completely. Think it is mostly the public that still holds this view.
I'm incline to believe the universe is infinite due to my inability to imagine "nothing" or logically get to "nothing." It seems to me, if "nothing" cannot exist then something must exist at all points, making the universe infinite. But this is just my own conclusion. Although I've also heard physicist say "the universe may be finite or infinite." So at least I'm not alone in this.