Finitism doesn’t reject the existence of any given natural number (although ultrafinitism might), nor the validity of the successor function (counting), nor even the notion of a “potential” infinity (like time), just the idea of a completed one being an object in its own right (which can be put into a set). The Axiom of Infinity doesn’t let you escape the notion of classes which can’t themselves be an element of a set. Set theory runs into paradoxes if we allow it. Is it such an invalid move to disallow the class of Naturals as an element of a set, when even ZFC must disallow the Surreals for similar reasons?
Before Cantor, all mathematicians were finitists. It’s not a weird position historically.
We do model physics with “real” numbers, but that doesn’t mean the underlying reality is infinite or even infinitely divisible. My finitism is motivated by my understanding of physics and cosmology, not the other way around. Nature seems to cut us off from any access to any completed infinity, and it’s not clear that even potential infinities are allowed (hence my sympathy with ultrafinitism). I have no need of that axiom.
Quantum Field Theory, though traditionally modeled using continuous mathematics, implies the Bekenstein bound: a finite region of space contains a finite amount of information. There are no “infinite bits” available to build the real numbers with. However densely you store information, eventually, at some point, your media collapses into a black hole, and packing in more must take up more space.
Physical space can’t be a continuum like the “reals”. It’s not infinitely divisible. Measuring distance with increasing precision requires higher frequency waves, and thus higher energies, which eventually has enough effective mass to gravitationally distort the very space you are measuring, eventually collapsing into a black hole.
Below a certain limit, distance isn’t physically meaningful. If you assume an electron is a point particle with “infinitesimal” size and you zoom in enough, you should be able to get arbitrarily high electric field strength. But at some point, high enough field strength results in vacuum polarization: virtual electron/positron pairs get pushed around and finally one of the positrons annihilates whatever you thought the real electron was, and then one of the virtual electrons doesn’t have anything to pair with and becomes the real one. It’s as if the electron is jumping around. You can’t nail it down. It doesn’t physically have a position down below a certain scale in time and space. There are no infinite bits. All the fundamental particle types are like this. There are no infinitesimal point particles. They’re just waves.
There’s also a cosmological horizon limiting how much of the Universe we can see. There’s also a (related) past temporal horizon at the Big Bang. We can’t see a completed past-temporal or spacial infinity, in any direction. We’re not sure of the Ultimate Fate of the Universe, but it looks like Heat Death is probably it, given our current understanding of physics. So there’s a future limit as well. The other likely candidate Fates are finite in time as well.
But even supposing finite information content in a finite region seems to be enough to make potential-infinite time not really meaningful. There’s a finite number of states possible, so eventually all reachable states are reached. If physics is deterministic (it seems to be), then we get into a cycle. So time is better modeled as a finite circle, rather than an infinite line. And if it’s not deterministic? Then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no phycial way to tell the difference.
Potential-infinite space is the same way. Any accessible region has a finite number of states, so at least some of them must repeat exactly in other regions. If there’s some determinism to the pattern, then it’s maybe better modeled as some curled-up finite space (although aperiodic tilings are also possible). If it’s random, then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no physical way to tell the difference. Once all reachable states have been saturated, why does it matter if they appear only once or a googol or infinity times?
I don’t quite understand how actual infinity differs from potential infinity in this context. Time in ToR is considered one of the dimensions of space. How can space be considered “potential infinity”? It subjectively looks like that to a forward-traveling observer. But usually we use the paradigm of objective reality, where everything is assumed to exist equally. Together with the past and the future, if we recall ToR again. Are we supposed to have a special case here, where we need to switch to the paradigm of subjective reality?
I am familiar with the idea that “the information that enables us to act best is true”, but it seems to me to be just a beautiful phrase, because in most cases, in order to develop a model that enables us to act best, we still have to be guided by “truth” in the old, ordinary sense. That is, we obtain some initial “atoms of truth” through experience, but later we have to take care of their logical consistency. And we are not quite right to call some high-level construction “truth”, even if it works well, if it does not logically agree with the “atoms” we used to create it.
This case is free from this problem, since practical verification in this area is impossible. But still, the feeling of some hypocrisy in front of oneself does not disappear. To admit at least in the edge of consciousness the possibility that “the Universe is not X” and at the same time use only “the Universe is X” in calculations—there is some kind of contradiction in this. This is either an act of doublethink (for an agnostic) or an act of politeness (for an ultrafinitist).
The difference between repeating patterns could manifest itself if there were interactions between them. But here reality rather speaks in favor of ultrafinitism. If the hierarchy of complication of structures with interactions could continue to infinity (even at the cost of slowing down the interactions), then theoretically we could find ourselves at any level of the hierarchy. Then we most likely would not see the “bottom of the hierarchy” (Planck’s limit). However, we see it. Therefore, either the hierarchy is finite, or something prevents interactions between levels, or some factor prevents the emergence of too high-level observers.
However, only the first option directly speaks in favor of ultrafinitism. The second and third options—like your reasoning—are valid only in the paradigm of subjective reality.
“Heat death” is also the end of time only in the paradigm of subjective reality. Moreover, only for an anthropocentrically minded observer, from whose point of view one state of “white noise” is no different from another.
Before Einstein, in the era of Newtonian ideas about time, it was believed that the magnitude of the past that had already taken place could be infinitely large. St. Augustine disagreed with this, but he had rather religious reasons.
Finitism doesn’t reject the existence of any given natural number (although ultrafinitism might), nor the validity of the successor function (counting), nor even the notion of a “potential” infinity (like time), just the idea of a completed one being an object in its own right (which can be put into a set). The Axiom of Infinity doesn’t let you escape the notion of classes which can’t themselves be an element of a set. Set theory runs into paradoxes if we allow it. Is it such an invalid move to disallow the class of Naturals as an element of a set, when even ZFC must disallow the Surreals for similar reasons?
Before Cantor, all mathematicians were finitists. It’s not a weird position historically.
We do model physics with “real” numbers, but that doesn’t mean the underlying reality is infinite or even infinitely divisible. My finitism is motivated by my understanding of physics and cosmology, not the other way around. Nature seems to cut us off from any access to any completed infinity, and it’s not clear that even potential infinities are allowed (hence my sympathy with ultrafinitism). I have no need of that axiom.
Quantum Field Theory, though traditionally modeled using continuous mathematics, implies the Bekenstein bound: a finite region of space contains a finite amount of information. There are no “infinite bits” available to build the real numbers with. However densely you store information, eventually, at some point, your media collapses into a black hole, and packing in more must take up more space.
Physical space can’t be a continuum like the “reals”. It’s not infinitely divisible. Measuring distance with increasing precision requires higher frequency waves, and thus higher energies, which eventually has enough effective mass to gravitationally distort the very space you are measuring, eventually collapsing into a black hole.
Below a certain limit, distance isn’t physically meaningful. If you assume an electron is a point particle with “infinitesimal” size and you zoom in enough, you should be able to get arbitrarily high electric field strength. But at some point, high enough field strength results in vacuum polarization: virtual electron/positron pairs get pushed around and finally one of the positrons annihilates whatever you thought the real electron was, and then one of the virtual electrons doesn’t have anything to pair with and becomes the real one. It’s as if the electron is jumping around. You can’t nail it down. It doesn’t physically have a position down below a certain scale in time and space. There are no infinite bits. All the fundamental particle types are like this. There are no infinitesimal point particles. They’re just waves.
There’s also a cosmological horizon limiting how much of the Universe we can see. There’s also a (related) past temporal horizon at the Big Bang. We can’t see a completed past-temporal or spacial infinity, in any direction. We’re not sure of the Ultimate Fate of the Universe, but it looks like Heat Death is probably it, given our current understanding of physics. So there’s a future limit as well. The other likely candidate Fates are finite in time as well.
But even supposing finite information content in a finite region seems to be enough to make potential-infinite time not really meaningful. There’s a finite number of states possible, so eventually all reachable states are reached. If physics is deterministic (it seems to be), then we get into a cycle. So time is better modeled as a finite circle, rather than an infinite line. And if it’s not deterministic? Then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no phycial way to tell the difference.
Potential-infinite space is the same way. Any accessible region has a finite number of states, so at least some of them must repeat exactly in other regions. If there’s some determinism to the pattern, then it’s maybe better modeled as some curled-up finite space (although aperiodic tilings are also possible). If it’s random, then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no physical way to tell the difference. Once all reachable states have been saturated, why does it matter if they appear only once or a googol or infinity times?
I don’t quite understand how actual infinity differs from potential infinity in this context. Time in ToR is considered one of the dimensions of space. How can space be considered “potential infinity”? It subjectively looks like that to a forward-traveling observer. But usually we use the paradigm of objective reality, where everything is assumed to exist equally. Together with the past and the future, if we recall ToR again. Are we supposed to have a special case here, where we need to switch to the paradigm of subjective reality?
I am familiar with the idea that “the information that enables us to act best is true”, but it seems to me to be just a beautiful phrase, because in most cases, in order to develop a model that enables us to act best, we still have to be guided by “truth” in the old, ordinary sense. That is, we obtain some initial “atoms of truth” through experience, but later we have to take care of their logical consistency. And we are not quite right to call some high-level construction “truth”, even if it works well, if it does not logically agree with the “atoms” we used to create it.
This case is free from this problem, since practical verification in this area is impossible. But still, the feeling of some hypocrisy in front of oneself does not disappear. To admit at least in the edge of consciousness the possibility that “the Universe is not X” and at the same time use only “the Universe is X” in calculations—there is some kind of contradiction in this. This is either an act of doublethink (for an agnostic) or an act of politeness (for an ultrafinitist).
The difference between repeating patterns could manifest itself if there were interactions between them. But here reality rather speaks in favor of ultrafinitism. If the hierarchy of complication of structures with interactions could continue to infinity (even at the cost of slowing down the interactions), then theoretically we could find ourselves at any level of the hierarchy. Then we most likely would not see the “bottom of the hierarchy” (Planck’s limit). However, we see it. Therefore, either the hierarchy is finite, or something prevents interactions between levels, or some factor prevents the emergence of too high-level observers.
However, only the first option directly speaks in favor of ultrafinitism. The second and third options—like your reasoning—are valid only in the paradigm of subjective reality.
“Heat death” is also the end of time only in the paradigm of subjective reality. Moreover, only for an anthropocentrically minded observer, from whose point of view one state of “white noise” is no different from another.
Before Einstein, in the era of Newtonian ideas about time, it was believed that the magnitude of the past that had already taken place could be infinitely large. St. Augustine disagreed with this, but he had rather religious reasons.