Do infinite dimensional systems make sense? [closed]2019 Community Moderator Election Unicorn Meta Zoo #1: Why another podcast? Announcing the arrival of Valued Associate #679: Cesar Manara 2019 Moderator Election Q&A - QuestionnaireNonseparable Hilbert spacePath integral vs. measure on infinite dimensional spaceQuantum computing and quantum controlExamples of discrete Hamiltonians?References on experimental realization of quantum one-dimensional infinite-well modelWhy do we need infinite-dimensional Hilbert spaces in physics?Good book for learning about mathematical foundation of quantum physicsKochen-Specker property in infinite dimensional systemsAre there fundamental differences between finite and infinite systems?Infinite-dimensional Hilbert spaces in QM vs. finite-dimensional Hilbert spaces in quantum gravity?Help me make sense of the spectrum for the quantum wave function of an infinitely hard equilateral triangle
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Do infinite dimensional systems make sense? [closed]
2019 Community Moderator Election
Unicorn Meta Zoo #1: Why another podcast?
Announcing the arrival of Valued Associate #679: Cesar Manara
2019 Moderator Election Q&A - QuestionnaireNonseparable Hilbert spacePath integral vs. measure on infinite dimensional spaceQuantum computing and quantum controlExamples of discrete Hamiltonians?References on experimental realization of quantum one-dimensional infinite-well modelWhy do we need infinite-dimensional Hilbert spaces in physics?Good book for learning about mathematical foundation of quantum physicsKochen-Specker property in infinite dimensional systemsAre there fundamental differences between finite and infinite systems?Infinite-dimensional Hilbert spaces in QM vs. finite-dimensional Hilbert spaces in quantum gravity?Help me make sense of the spectrum for the quantum wave function of an infinitely hard equilateral triangle
$begingroup$
I'm learning infinite-dimensional systems in mathematical viewpoint and trying to understand it from physical perspective.
I would like to understand if infinite-dimensional systems make sense in physics, especially when it becomes necessary in quantum control theory. Are there any simple and intuitive examples?
quantum-mechanics hilbert-space
edited Apr 5 at 12:00
Ruslan
9,90243175
asked Apr 5 at 9:36
GaoGao
364
$endgroup$
closed as primarily opinion-based by GiorgioP, ZeroTheHero, FGSUZ, Aaron Stevens, Jon Custer Apr 8 at 15:16
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I'm learning infinite-dimensional systems in mathematical viewpoint and trying to understand it from physical perspective.
I would like to understand if infinite-dimensional systems make sense in physics, especially when it becomes necessary in quantum control theory. Are there any simple and intuitive examples?
quantum-mechanics hilbert-space
edited Apr 5 at 12:00
Ruslan
9,90243175
asked Apr 5 at 9:36
GaoGao
364
$endgroup$
closed as primarily opinion-based by GiorgioP, ZeroTheHero, FGSUZ, Aaron Stevens, Jon Custer Apr 8 at 15:16
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
"Makes sense" is kind of hard to interpret. But they're definitely useful as models describing the observable behavior of some systems. In fact, you're probably referring to countably-infinite-dimensional spaces, aka separable. There are also uses for uncountably-infinite-dimensional (aka non separable) spaces, e.g., physics.stackexchange.com/q/60608 (and google for lots more stuff).
$endgroup$
– John Forkosh
Apr 5 at 10:56
add a comment |
$begingroup$
I'm learning infinite-dimensional systems in mathematical viewpoint and trying to understand it from physical perspective.
I would like to understand if infinite-dimensional systems make sense in physics, especially when it becomes necessary in quantum control theory. Are there any simple and intuitive examples?
quantum-mechanics hilbert-space
edited Apr 5 at 12:00
Ruslan
9,90243175
asked Apr 5 at 9:36
GaoGao
364
$endgroup$
I'm learning infinite-dimensional systems in mathematical viewpoint and trying to understand it from physical perspective.
I would like to understand if infinite-dimensional systems make sense in physics, especially when it becomes necessary in quantum control theory. Are there any simple and intuitive examples?
quantum-mechanics hilbert-space
quantum-mechanics hilbert-space
edited Apr 5 at 12:00
Ruslan
9,90243175
asked Apr 5 at 9:36
GaoGao
364
edited Apr 5 at 12:00
Ruslan
9,90243175
asked Apr 5 at 9:36
GaoGao
364
edited Apr 5 at 12:00
Ruslan
9,90243175
edited Apr 5 at 12:00
Ruslan
9,90243175
edited Apr 5 at 12:00
Ruslan
9,90243175
9,90243175
asked Apr 5 at 9:36
GaoGao
364
asked Apr 5 at 9:36
GaoGao
364
asked Apr 5 at 9:36
GaoGao
364
364
closed as primarily opinion-based by GiorgioP, ZeroTheHero, FGSUZ, Aaron Stevens, Jon Custer Apr 8 at 15:16
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as primarily opinion-based by GiorgioP, ZeroTheHero, FGSUZ, Aaron Stevens, Jon Custer Apr 8 at 15:16
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
"Makes sense" is kind of hard to interpret. But they're definitely useful as models describing the observable behavior of some systems. In fact, you're probably referring to countably-infinite-dimensional spaces, aka separable. There are also uses for uncountably-infinite-dimensional (aka non separable) spaces, e.g., physics.stackexchange.com/q/60608 (and google for lots more stuff).
$endgroup$
– John Forkosh
Apr 5 at 10:56
add a comment |
5
$begingroup$
"Makes sense" is kind of hard to interpret. But they're definitely useful as models describing the observable behavior of some systems. In fact, you're probably referring to countably-infinite-dimensional spaces, aka separable. There are also uses for uncountably-infinite-dimensional (aka non separable) spaces, e.g., physics.stackexchange.com/q/60608 (and google for lots more stuff).
$endgroup$
– John Forkosh
Apr 5 at 10:56
5
5
$begingroup$
"Makes sense" is kind of hard to interpret. But they're definitely useful as models describing the observable behavior of some systems. In fact, you're probably referring to countably-infinite-dimensional spaces, aka separable. There are also uses for uncountably-infinite-dimensional (aka non separable) spaces, e.g., physics.stackexchange.com/q/60608 (and google for lots more stuff).
$endgroup$
– John Forkosh
Apr 5 at 10:56
$begingroup$
"Makes sense" is kind of hard to interpret. But they're definitely useful as models describing the observable behavior of some systems. In fact, you're probably referring to countably-infinite-dimensional spaces, aka separable. There are also uses for uncountably-infinite-dimensional (aka non separable) spaces, e.g., physics.stackexchange.com/q/60608 (and google for lots more stuff).
$endgroup$
– John Forkosh
Apr 5 at 10:56
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Welcome to Stack Exchange!
I do not know much about quantum control theory, but I can give you a simple example from regular quantum mechanics: that of a particle in a box. This is one of the simplest systems one can study in QM, but even here an infinite dimensional space shows up.
Indexing the energy eigenstates by $n$ so that $$H|nrangle=E_n|nrangle$$ there is an infinite number of possible states, one for every integer. Every state with its own energy: $$E_n=fracn^2pi^2hbar^22mL^2.$$ Thus if you want to describe a general quantum state in this system you would write it down as $$|psirangle=sum_n=1^infty c_n|nrangle,$$ where $c_n$ is a complex number. $|psirangle$ is then an example of a vector in an infinite dimensional space where every possible state is a basis vector, and the $c_n$ are the expansion coefficients in that basis.
You can of course imagine the $n$ indexing some other collection of states of some other system. Indeed, in most cases the dimension of the space of all possible states of a quantum system will be infinite dimensional.
answered Apr 5 at 10:56
JSorngardJSorngard
37918
$endgroup$
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
add a comment |
$begingroup$
"Infinite" for me does not make much sense in physics. It is a nice mathematical tools, you need it to make calculations; but for real things I prefer "very large". I have seen many "very large" things; I have never seen anything infinite.
Anyway, more seriously, we use all the time infinite dimensional spaces, they are useful. From the example of "particle in a box" by >JSorngard:
You want to keep your particle in one eigenstate, against external disturbance. It can excape going to other eigenstates. Those, in theory, are infinite, so to calculate the probability for your particle to fall off from your favourite eigenstate, you sum the transition probability to each of all those (infinite) states. And it works!!
In practice there are not really infinite eigenstates; the particle is confined by some actual physical trap that is finite in size and can hold only finite energy. But incredibly often you can disregard this finiteness, as the infinite sum is almost identical to the (very large) sum of the actual eigenstate.
Another argument in favour of usefulness/reality of infinites is about notation: we use infinite things to define & manipulate a normal object.
Example is the Taylor expansion of $e^x$:
It is an infinite sum, its useful, don't give rise to anything nonsensical.
answered Apr 5 at 13:44
pattapatta
77648
$endgroup$
2
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
1
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
2
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
1
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
|
show 1 more comment
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Welcome to Stack Exchange!
I do not know much about quantum control theory, but I can give you a simple example from regular quantum mechanics: that of a particle in a box. This is one of the simplest systems one can study in QM, but even here an infinite dimensional space shows up.
Indexing the energy eigenstates by $n$ so that $$H|nrangle=E_n|nrangle$$ there is an infinite number of possible states, one for every integer. Every state with its own energy: $$E_n=fracn^2pi^2hbar^22mL^2.$$ Thus if you want to describe a general quantum state in this system you would write it down as $$|psirangle=sum_n=1^infty c_n|nrangle,$$ where $c_n$ is a complex number. $|psirangle$ is then an example of a vector in an infinite dimensional space where every possible state is a basis vector, and the $c_n$ are the expansion coefficients in that basis.
You can of course imagine the $n$ indexing some other collection of states of some other system. Indeed, in most cases the dimension of the space of all possible states of a quantum system will be infinite dimensional.
answered Apr 5 at 10:56
JSorngardJSorngard
37918
$endgroup$
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
add a comment |
$begingroup$
Welcome to Stack Exchange!
I do not know much about quantum control theory, but I can give you a simple example from regular quantum mechanics: that of a particle in a box. This is one of the simplest systems one can study in QM, but even here an infinite dimensional space shows up.
Indexing the energy eigenstates by $n$ so that $$H|nrangle=E_n|nrangle$$ there is an infinite number of possible states, one for every integer. Every state with its own energy: $$E_n=fracn^2pi^2hbar^22mL^2.$$ Thus if you want to describe a general quantum state in this system you would write it down as $$|psirangle=sum_n=1^infty c_n|nrangle,$$ where $c_n$ is a complex number. $|psirangle$ is then an example of a vector in an infinite dimensional space where every possible state is a basis vector, and the $c_n$ are the expansion coefficients in that basis.
You can of course imagine the $n$ indexing some other collection of states of some other system. Indeed, in most cases the dimension of the space of all possible states of a quantum system will be infinite dimensional.
answered Apr 5 at 10:56
JSorngardJSorngard
37918
$endgroup$
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
add a comment |
$begingroup$
Welcome to Stack Exchange!
I do not know much about quantum control theory, but I can give you a simple example from regular quantum mechanics: that of a particle in a box. This is one of the simplest systems one can study in QM, but even here an infinite dimensional space shows up.
Indexing the energy eigenstates by $n$ so that $$H|nrangle=E_n|nrangle$$ there is an infinite number of possible states, one for every integer. Every state with its own energy: $$E_n=fracn^2pi^2hbar^22mL^2.$$ Thus if you want to describe a general quantum state in this system you would write it down as $$|psirangle=sum_n=1^infty c_n|nrangle,$$ where $c_n$ is a complex number. $|psirangle$ is then an example of a vector in an infinite dimensional space where every possible state is a basis vector, and the $c_n$ are the expansion coefficients in that basis.
You can of course imagine the $n$ indexing some other collection of states of some other system. Indeed, in most cases the dimension of the space of all possible states of a quantum system will be infinite dimensional.
answered Apr 5 at 10:56
JSorngardJSorngard
37918
$endgroup$
Welcome to Stack Exchange!
I do not know much about quantum control theory, but I can give you a simple example from regular quantum mechanics: that of a particle in a box. This is one of the simplest systems one can study in QM, but even here an infinite dimensional space shows up.
Indexing the energy eigenstates by $n$ so that $$H|nrangle=E_n|nrangle$$ there is an infinite number of possible states, one for every integer. Every state with its own energy: $$E_n=fracn^2pi^2hbar^22mL^2.$$ Thus if you want to describe a general quantum state in this system you would write it down as $$|psirangle=sum_n=1^infty c_n|nrangle,$$ where $c_n$ is a complex number. $|psirangle$ is then an example of a vector in an infinite dimensional space where every possible state is a basis vector, and the $c_n$ are the expansion coefficients in that basis.
You can of course imagine the $n$ indexing some other collection of states of some other system. Indeed, in most cases the dimension of the space of all possible states of a quantum system will be infinite dimensional.
answered Apr 5 at 10:56
JSorngardJSorngard
37918
edited Apr 5 at 11:09
answered Apr 5 at 10:56
JSorngardJSorngard
37918
answered Apr 5 at 10:56
JSorngardJSorngard
37918
answered Apr 5 at 10:56
JSorngardJSorngard
37918
37918
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
add a comment |
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
$begingroup$
Welcome on the Stack Exchange :-)
$endgroup$
– peterh
Apr 6 at 11:06
add a comment |
$begingroup$
"Infinite" for me does not make much sense in physics. It is a nice mathematical tools, you need it to make calculations; but for real things I prefer "very large". I have seen many "very large" things; I have never seen anything infinite.
Anyway, more seriously, we use all the time infinite dimensional spaces, they are useful. From the example of "particle in a box" by >JSorngard:
You want to keep your particle in one eigenstate, against external disturbance. It can excape going to other eigenstates. Those, in theory, are infinite, so to calculate the probability for your particle to fall off from your favourite eigenstate, you sum the transition probability to each of all those (infinite) states. And it works!!
In practice there are not really infinite eigenstates; the particle is confined by some actual physical trap that is finite in size and can hold only finite energy. But incredibly often you can disregard this finiteness, as the infinite sum is almost identical to the (very large) sum of the actual eigenstate.
Another argument in favour of usefulness/reality of infinites is about notation: we use infinite things to define & manipulate a normal object.
Example is the Taylor expansion of $e^x$:
It is an infinite sum, its useful, don't give rise to anything nonsensical.
answered Apr 5 at 13:44
pattapatta
77648
$endgroup$
2
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
1
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
2
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
1
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
|
show 1 more comment
$begingroup$
"Infinite" for me does not make much sense in physics. It is a nice mathematical tools, you need it to make calculations; but for real things I prefer "very large". I have seen many "very large" things; I have never seen anything infinite.
Anyway, more seriously, we use all the time infinite dimensional spaces, they are useful. From the example of "particle in a box" by >JSorngard:
You want to keep your particle in one eigenstate, against external disturbance. It can excape going to other eigenstates. Those, in theory, are infinite, so to calculate the probability for your particle to fall off from your favourite eigenstate, you sum the transition probability to each of all those (infinite) states. And it works!!
In practice there are not really infinite eigenstates; the particle is confined by some actual physical trap that is finite in size and can hold only finite energy. But incredibly often you can disregard this finiteness, as the infinite sum is almost identical to the (very large) sum of the actual eigenstate.
Another argument in favour of usefulness/reality of infinites is about notation: we use infinite things to define & manipulate a normal object.
Example is the Taylor expansion of $e^x$:
It is an infinite sum, its useful, don't give rise to anything nonsensical.
answered Apr 5 at 13:44
pattapatta
77648
$endgroup$
2
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
1
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
2
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
1
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
|
show 1 more comment
$begingroup$
"Infinite" for me does not make much sense in physics. It is a nice mathematical tools, you need it to make calculations; but for real things I prefer "very large". I have seen many "very large" things; I have never seen anything infinite.
Anyway, more seriously, we use all the time infinite dimensional spaces, they are useful. From the example of "particle in a box" by >JSorngard:
You want to keep your particle in one eigenstate, against external disturbance. It can excape going to other eigenstates. Those, in theory, are infinite, so to calculate the probability for your particle to fall off from your favourite eigenstate, you sum the transition probability to each of all those (infinite) states. And it works!!
In practice there are not really infinite eigenstates; the particle is confined by some actual physical trap that is finite in size and can hold only finite energy. But incredibly often you can disregard this finiteness, as the infinite sum is almost identical to the (very large) sum of the actual eigenstate.
Another argument in favour of usefulness/reality of infinites is about notation: we use infinite things to define & manipulate a normal object.
Example is the Taylor expansion of $e^x$:
It is an infinite sum, its useful, don't give rise to anything nonsensical.
answered Apr 5 at 13:44
pattapatta
77648
$endgroup$
"Infinite" for me does not make much sense in physics. It is a nice mathematical tools, you need it to make calculations; but for real things I prefer "very large". I have seen many "very large" things; I have never seen anything infinite.
Anyway, more seriously, we use all the time infinite dimensional spaces, they are useful. From the example of "particle in a box" by >JSorngard:
You want to keep your particle in one eigenstate, against external disturbance. It can excape going to other eigenstates. Those, in theory, are infinite, so to calculate the probability for your particle to fall off from your favourite eigenstate, you sum the transition probability to each of all those (infinite) states. And it works!!
In practice there are not really infinite eigenstates; the particle is confined by some actual physical trap that is finite in size and can hold only finite energy. But incredibly often you can disregard this finiteness, as the infinite sum is almost identical to the (very large) sum of the actual eigenstate.
Another argument in favour of usefulness/reality of infinites is about notation: we use infinite things to define & manipulate a normal object.
Example is the Taylor expansion of $e^x$:
It is an infinite sum, its useful, don't give rise to anything nonsensical.
answered Apr 5 at 13:44
pattapatta
77648
answered Apr 5 at 13:44
pattapatta
77648
answered Apr 5 at 13:44
pattapatta
77648
answered Apr 5 at 13:44
pattapatta
77648
77648
2
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
1
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
2
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
1
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
|
show 1 more comment
2
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
1
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
2
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
1
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
2
2
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
$begingroup$
"I have never seen anything infinite." are you sure?
$endgroup$
– Orangesandlemons
Apr 5 at 14:02
1
1
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
Well, ok, maybe I've seen it, but I didn't manage to see it all, my small brain recorded only a small part!
$endgroup$
– patta
Apr 5 at 14:16
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
$begingroup$
You will not "see" irrational numbers either, so I guess you can only use finite mathematics to do physics?
$endgroup$
– Martin Argerami
Apr 5 at 14:20
2
2
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
$begingroup$
Yes, my computer and all sensors I know use only integers; irrationals are just approximated with large integers. About brain and analog machines, we can discuss... My meaning was that a tool (infinity, irrationals..) can "make few sense" in physics, while being actually useful.
$endgroup$
– patta
Apr 5 at 14:22
1
1
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
$begingroup$
Do you happen, by chance, to be a finitist?
$endgroup$
– Don Thousand
Apr 6 at 13:12
|
show 1 more comment
5
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"Makes sense" is kind of hard to interpret. But they're definitely useful as models describing the observable behavior of some systems. In fact, you're probably referring to countably-infinite-dimensional spaces, aka separable. There are also uses for uncountably-infinite-dimensional (aka non separable) spaces, e.g., physics.stackexchange.com/q/60608 (and google for lots more stuff).
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– John Forkosh
Apr 5 at 10:56