The physics of infinity(nature.com)
nature.com
The physics of infinity
https://www.nature.com/articles/s41567-018-0238-1.epdf?shared_access_token=nhIyZJldj4QzWZs7LvZIttRgN0jAjWel9jnR3ZoTv0PMOaEWTfe0Iq_Ol3Eo9bd6Lh9xyPK-ya44kxWDxYi4IQo2Zqj-Ymd6yZVANNbW9FXmT1HwoVMnEtM00qpXT48gLDqpQXX3mvS3gRH22aRhLs-Cf_4dd6NkVcLZZP3rPbg%3D
99 comments
I don't get the dual argument about zero, for which the impossibility of a completely empty region of spacetime is given as an example. That may be so, but a region empty of countable things like atoms seems physically conceivable. I am writing from a room containing zero elephants.
Rules of the macroscopic world do not apply to the subatomic (or even molecular) world. There are no two identical elephants in the world. On the other hand, you will have a hard time to tell two electrons or even two molecules of the same configuration apart from another.
A total empty region, void of any atoms etc., still contains 'quantum noise' and thus virtual particles. The world, the universe as we know it so far has become inconceivable without this observation, known as the uncertainty principle.
I find the argument quite convincing.
A total empty region, void of any atoms etc., still contains 'quantum noise' and thus virtual particles. The world, the universe as we know it so far has become inconceivable without this observation, known as the uncertainty principle.
I find the argument quite convincing.
> The world, the universe as we know it so far has become inconceivable without this observation, known as the uncertainty principle.
Not really, deterministic interpretations of QM don't require such mental contortions.
Not really, deterministic interpretations of QM don't require such mental contortions.
[deleted]
I'm far from an expert but any deterministic interpretation of QM (including the popular De Broglie–Bohm) is required to still maintain the uncertainty principle - otherwise it would be falsified by showing it conflicts with the results of hundreds of known experiments.
Yes, but it violates the original claim that rules of the macroscopic world don't apply in this realm. de Broglie-Bohm is a classical theory with a extra term to account for quantum behaviour.
The field theory covering "quantum noise", the existence of virtual particles, and the vacuum is also completely different.
The field theory covering "quantum noise", the existence of virtual particles, and the vacuum is also completely different.
[deleted]
Old shaky hands aiming a looking glass at one of two moving elephants standing close by each other would tell a different story, perhaps.
I'm not sure how well established Virtual Quantum Noise is, but saying that an empty place contains this is just wrong. If it has an effect in the propagation of a real particle in to said space, it wouldn't be empty anymore. That's confusing, and it might be helpful to note that these quantum fluctuations happen on the boundaries of the space, not just everywhere (and if they did, they wouldn't matter any more than my imaginary friend the pink elephant Safr riding the fluctuations).
You are talking of black holes, by the way, aren't you?
I'm not sure how well established Virtual Quantum Noise is, but saying that an empty place contains this is just wrong. If it has an effect in the propagation of a real particle in to said space, it wouldn't be empty anymore. That's confusing, and it might be helpful to note that these quantum fluctuations happen on the boundaries of the space, not just everywhere (and if they did, they wouldn't matter any more than my imaginary friend the pink elephant Safr riding the fluctuations).
You are talking of black holes, by the way, aren't you?
[deleted]
The complete wave function of your room contains some elephant-like fluctuations in the distant decimal places.
A functioning brain is posited to spontaneously appear in about 10^10^50 years due to random fluctuations.
https://en.wikipedia.org/wiki/Boltzmann_brain
I'm not sure if a functioning elephant is more or less likely to be honest....
https://en.wikipedia.org/wiki/Boltzmann_brain
I'm not sure if a functioning elephant is more or less likely to be honest....
I have met the Queen of England exactly zero times.
To piggyback, and inflect on, whatshisface's point, the number of times you've met the Queen of England is not a physical thing. You can't describe the Queen of England with a physical function either, even if you could describe her physical body, you can't capture what she means to the United Kingdom, which is at least 40% of the context being dealt with here.
And the QoE isn't a discrete entity. Her molecules are constantly reacting and atoms are entering and leaving her. One h2o is some distant decimal place percentage of the QoE
The complete wavefunction of you and the Queen of England involves a handshake-like fluctuation in the distant decimal places.
But once either of us dies there won't. Zero can be meaningful in describing the world. I agree infinity, in its truest sense, can't... but zero definitely can.
No, no, null is not meaningful. That's the whole point. It is meaningful as neutral element of a ring, as center of a coordinate system or as limit of some function, sure, but only in comparison to other values and the other values are all that matters. Not having the other values, having literally nothing, the word ''having'' would be meaningless. There is always something.
There's something about nihilism. It's being a self defeating argument, but educating. Then there's constructivism in various senses.
There's something about finitism, too, computable numbers and what not. Infinity is rather ... monotonous.
There's something about nihilism. It's being a self defeating argument, but educating. Then there's constructivism in various senses.
There's something about finitism, too, computable numbers and what not. Infinity is rather ... monotonous.
but how is 0 any different from any other number then?
If you're going to look at more than just integers, infinity is everywhere, in between every number you can count, and then some! 0 is as meaningless but so is every other number!
If you limit to integers or even just whole numbers, 0 is meaningful, just like all the numbers...
If you're going to look at more than just integers, infinity is everywhere, in between every number you can count, and then some! 0 is as meaningless but so is every other number!
If you limit to integers or even just whole numbers, 0 is meaningful, just like all the numbers...
0 is a symbol, but that symbol has quite literally no value. It is special that way.
What's the rest mass of a photon? What's the electric charge of a neutrino? What's the spin of a Higgs boson?
> region empty of countable things like atoms seems physically conceivable
As far as I know this is not physically conceivable, if physically conceivable means physical laws as we understand them.
Quantum vacuum state, or zero-point field is not completely void empty.
As far as I know this is not physically conceivable, if physically conceivable means physical laws as we understand them.
Quantum vacuum state, or zero-point field is not completely void empty.
My [lay] understanding is that a portion of spacetime has yet to be found that has 0 countable things. I also believe 'countable' gets tricky the more you zoom in, as it were.
I think that's the craziest thing, every time we get smaller, we find more stuff. "Atom" means "indivisible". It was supposed to be the smallest unit of matter.
Then we found things smaller. Sub-atomic particles. Literally makes no sense because it literally means smaller than the smallest thing.
The universe is apparently fractal af.
Then we found things smaller. Sub-atomic particles. Literally makes no sense because it literally means smaller than the smallest thing.
The universe is apparently fractal af.
Your room and elephants are different objects. Your room has some size and it is not empty but it has zero elephants or a finite number of zero size elephants.
0 = nothing = non-existence.
0 = nothing = non-existence.
How would you eliminate electromagnetic fields or inconceivably small particles like neutrinos from entering?
From the article:
> There is a duality between zero and infinity, expressed in the elementary identity 1/0 = ∞.
I understand what they're saying here - but I was always under the impression that, rather than yielding infinity, division by zero was undefined. Wikipedia [1] says that this is true "In ordinary arithmetic", but goes on to say that it is "sometimes useful" to think of a "formal calculation" involving division by zero as evaluating to infinity. A formal calculation seems to be one that ignores whether the result is well defined.
Would anyone care to comment on whether the mathematical perspective above carries over into the physics domain?
[1] https://en.wikipedia.org/wiki/Division_by_zero
> There is a duality between zero and infinity, expressed in the elementary identity 1/0 = ∞.
I understand what they're saying here - but I was always under the impression that, rather than yielding infinity, division by zero was undefined. Wikipedia [1] says that this is true "In ordinary arithmetic", but goes on to say that it is "sometimes useful" to think of a "formal calculation" involving division by zero as evaluating to infinity. A formal calculation seems to be one that ignores whether the result is well defined.
Would anyone care to comment on whether the mathematical perspective above carries over into the physics domain?
[1] https://en.wikipedia.org/wiki/Division_by_zero
Division by 0 is not possible in fields [1], which are the algebraic structure most commonly associated with “ordinary arithmetic.” For example, the real numbers comprise an ordered and complete field. You cannot divide by 0 in a field essentially by definition, so it’s undefined.
However, you can define meaningful algebras in which division by 0 is not only trivially allowed, but also equal to infinity. This is possible by extending the complexes, for example. More generally, see [2].
Note I’m speaking strictly of the math here. I can’t comment on this usage in physics specifically. But there’s no issue with defining 1/0 = ∞ if the rest of the theory remains fundamentally consistent.
______
1. https://en.m.wikipedia.org/wiki/Field_(mathematics)
2. https://en.m.wikipedia.org/wiki/Wheel_theory
However, you can define meaningful algebras in which division by 0 is not only trivially allowed, but also equal to infinity. This is possible by extending the complexes, for example. More generally, see [2].
Note I’m speaking strictly of the math here. I can’t comment on this usage in physics specifically. But there’s no issue with defining 1/0 = ∞ if the rest of the theory remains fundamentally consistent.
______
1. https://en.m.wikipedia.org/wiki/Field_(mathematics)
2. https://en.m.wikipedia.org/wiki/Wheel_theory
I disagree, only because infinity is not an element of the real or the complex numbers. You can make extended number systems (the hyperreals, for instance) but it isn't commonly done. Infinity as a notation in the limit is not the same thing as infinity as an element of a set.
Can you clarify what you're disagreeing with? I don't think I stated anything controversial.
No, ∞ is not an element of the real or complex fields, that's correct. But it is an element of the extended real and complex number systems, according to the following rules:
In particular, note that by augmenting ℝ with +∞, -∞, we can derive (x / 0) = ∞ from the third rule. This behavior is only very slightly pathological because 1) it's (mostly) contained to the infinities, and 2) it still mostly preserves the order, completeness, addition and multiplication properties of the real numbers themselves. It's just not a field because we cannot do some propositional things that follow from Peano arithmetic (e.g. cancellation) and we can't e.g. take square roots of infinity.
To make this point even more technical, we can choose to define numbers as cardinalities of sets under Zermelo-Fraenkel set theory instead of under Peano arithmetic. This allows us to neatly define arithmetic involving infinities via operations on infinite sets.
No, ∞ is not an element of the real or complex fields, that's correct. But it is an element of the extended real and complex number systems, according to the following rules:
for all x ∈ ℝ:
1) - ∞ < x < + ∞,
2) x + ∞ = + ∞, x - ∞ = - ∞,
3) (x / + ∞) = (x / - ∞) = 0,
4) x > 0 ==> x(+ ∞) = + ∞, x(- ∞) = - ∞,
5) x < 0 ==> x(+ ∞) = - ∞, x(- ∞) = + ∞
In particular, note that by augmenting ℝ with +∞, -∞, we can derive (x / 0) = ∞ from the third rule. This behavior is only very slightly pathological because 1) it's (mostly) contained to the infinities, and 2) it still mostly preserves the order, completeness, addition and multiplication properties of the real numbers themselves. It's just not a field because we cannot do some propositional things that follow from Peano arithmetic (e.g. cancellation) and we can't e.g. take square roots of infinity.
To make this point even more technical, we can choose to define numbers as cardinalities of sets under Zermelo-Fraenkel set theory instead of under Peano arithmetic. This allows us to neatly define arithmetic involving infinities via operations on infinite sets.
Upon rereading what you are saying, I realize I am not disagreeing with you. I am, however, easily triggered by this as an armchair student of math, because I think infinity forms a kind of a trap for amateurs like me. "Actual" infinity is not a member of the reals and it is not necessary to augment the reals to have a functioning real analysis. But you are right that it can be done, it can be added to the reals to form interesting algebraic structures and there are even books on nonstandard analysis that start from here (Elementary Calculus, which I have read part of but not finished).
Division by zero can certainly be given meaning in other mathematical contexts. The real projective line is one example [0]. There are other algebraic structures in ring theory that also have a division by zero, for example a ring with only a zero element. So I would say it depends on your context if something makes sense to do. You aren't ignoring or doing something ill defined, its the opposite (in mathematics). You can do it if its consistent mathematics.
Physicists have very good intuition but sometimes making their statements mathematically rigorous requires extra work. I think a good example is in the history of field theory. Over time the intuition was made into something more formal. The article in the OP discusses renormalisation.
[0] https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
Physicists have very good intuition but sometimes making their statements mathematically rigorous requires extra work. I think a good example is in the history of field theory. Over time the intuition was made into something more formal. The article in the OP discusses renormalisation.
[0] https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
You could think of approaching this result by dividing 1 by an ever smaller and smaller number and the result is getting bigger and bigger, with no end in sight, as you can always come up with a smaller number, never reaching 0.
There are several places in physics where you encounter infinities, the most prominent case probably is the aforementioned uncertainty principle, which leads to 'loops' in QED and need to be countered by renormalization[0].
[0] https://en.wikipedia.org/wiki/Renormalization
There are several places in physics where you encounter infinities, the most prominent case probably is the aforementioned uncertainty principle, which leads to 'loops' in QED and need to be countered by renormalization[0].
[0] https://en.wikipedia.org/wiki/Renormalization
You could say that is impossible to start from something finite and completely destroy it to 0, and this analogous to the impossibility of creating infinite pieces. But another option is that there is simply nothing to start with.
It's been a long time since I've done any physics and there is absolutely no rigour here, but the (extremely useful) Dirac delta function comes to mind. You can think of it as being infinitely tall and infinitely thin, but with unit area. If you tried to divide the "area" (1) by the "width" (0), you'd get the height (infinity).
~$ node
> 1/0
Infinity
Q.E.D.
unless you want to claim that javascript is occasionaly illdefined.
> 1/0
Infinity
Q.E.D.
unless you want to claim that javascript is occasionaly illdefined.
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the result is well defined in terms of a limit: lim 1/x as x -> 0 = ∞. I consider 1/0 = ∞ a practical shortcut notation.
Except it’s also negative infinity.
Actually, ∞. +∞.and -∞ are three different things.
∞ and +∞ are the same thing afaik.
Technically ∞ is an element of the projective real line: R U {∞}. +∞ and -∞ are distinct elements in the extended real number system: R U {-∞, +∞}.
It depends on the context. ∞ is sometimes defined as the point on the projective real line equal to both +∞ and -∞.
Oh, right!
lim (x->0⁺) 1/x = ∞
lim (x->0⁺) 1/x = ∞
It is 0/0 that is undefined. For 1/0 the standard is either infinity (easily checkable in a language like JavaScript or C) or you get the usual talk that you can't mix infinity with numbers and the operation would be meaningless, etc.
The more sensible question, IMO, is "Is the universe fundamentally quantized?". Can you concretely say where the boundary between an apple and the surrounding air is? Is a particular electron on the boundary part of the apple or the air?
If the universe is not truly quantized, then there is nothing to count. By extension, mathematics is not related to physical reality, and it makes no sense to apply mathematical notions of infinity to the universe in such speculative ways.
If the universe is not truly quantized, then there is nothing to count. By extension, mathematics is not related to physical reality, and it makes no sense to apply mathematical notions of infinity to the universe in such speculative ways.
Even if the universe is not discretized, there is no reason to completely discount the value of mathematics. Continuous probability distributions of particle locations still have meaning, individual entities can still be disambiguated as local maxima of density gradients, counting can still be built up from raw axioms and set theory.
Even for people don't accept the "monism" (to use a metaphor to the monism/ dualism debate of the human mind and body) of physics and mathematics, these same people cannot avoid accepting that correctly applied theoretical manipulations of mathematical symbols jive perfectly well with the predicted, physical outcome they represent. Does claiming that the numbers don't mean anything mean anything?
Even for people don't accept the "monism" (to use a metaphor to the monism/ dualism debate of the human mind and body) of physics and mathematics, these same people cannot avoid accepting that correctly applied theoretical manipulations of mathematical symbols jive perfectly well with the predicted, physical outcome they represent. Does claiming that the numbers don't mean anything mean anything?
I don't know where you read that I "completely discount the value of mathematics".
>"...cannot avoid accepting that correctly applied theoretical manipulations of mathematical symbols jive perfectly well with the predicted, physical outcome they represent."
Close, except for "perfectly". Mathematics and other scientific models are approximations of reality and nothing more. They are fantastic tools but don't be misled into thinking they give a "perfect" picture.
>"...cannot avoid accepting that correctly applied theoretical manipulations of mathematical symbols jive perfectly well with the predicted, physical outcome they represent."
Close, except for "perfectly". Mathematics and other scientific models are approximations of reality and nothing more. They are fantastic tools but don't be misled into thinking they give a "perfect" picture.
Well said, I wish I knew you in real life.
> If the universe is not truly quantized... mathematics is not related to physical reality
How does that follow? Mathematics is perfectly capable of modeling continuous quantities.
How does that follow? Mathematics is perfectly capable of modeling continuous quantities.
Yes, math can model "continuous quantities", but is the idea of "quantity" anything more than a human abstraction? How do you define quantity, and can you do it without the notion of counting and discrete objects?
> is the idea of "quantity" anything more than a human abstraction?
What, to you, would not qualify as a human abstraction?
> How do you define quantity, and can you do it without the notion of counting and discrete objects?
Absolutely. See Tarski's axiomatization of the real numbers, for instance.
What, to you, would not qualify as a human abstraction?
> How do you define quantity, and can you do it without the notion of counting and discrete objects?
Absolutely. See Tarski's axiomatization of the real numbers, for instance.
> What, to you, would not qualify as a human abstraction?
Nothing insofar as "what" implies a load of human abstractions. But it's clear that there is a lot of "stuff" out there that no human has ever experienced. It is extremely improbable that humans exist at a "Goldilock's scale" wherein we are even physically capable of experiencing "everything" and of finding boundaries on the universe.
> Absolutely. See Tarski's axiomatization of the real numbers, for instance.
I meant counting in the broadest sense, which is where numbers themselves emerge from. It seems plausible that there are alien modes of cognition that don't rely on the notion of object and can approach continuous "stuff" more directly. Maybe even some terrestrial organisms work this way.
Nothing insofar as "what" implies a load of human abstractions. But it's clear that there is a lot of "stuff" out there that no human has ever experienced. It is extremely improbable that humans exist at a "Goldilock's scale" wherein we are even physically capable of experiencing "everything" and of finding boundaries on the universe.
> Absolutely. See Tarski's axiomatization of the real numbers, for instance.
I meant counting in the broadest sense, which is where numbers themselves emerge from. It seems plausible that there are alien modes of cognition that don't rely on the notion of object and can approach continuous "stuff" more directly. Maybe even some terrestrial organisms work this way.
> I meant counting in the broadest sense, which is where numbers themselves emerge from.
Are you asking how define quantity without quantity?
Also, I think it's fair to say there are quantities that no human has ever "experienced".
Are you asking how define quantity without quantity?
Also, I think it's fair to say there are quantities that no human has ever "experienced".
> David Hilbert famously argued that infinity cannot exist in physical reality.
This statement reduces the problem to the definitions of "existence" and "physical reality" which of course may have quite different interpretations. For example, the existence of particles (with finite mass and finite coordinates which can be "touched") and the existence of waves are rather different notions.
This statement reduces the problem to the definitions of "existence" and "physical reality" which of course may have quite different interpretations. For example, the existence of particles (with finite mass and finite coordinates which can be "touched") and the existence of waves are rather different notions.
It seems to me that something can only have existence for us if it can have a transitive causal influence upon us, and surely both particles and waves exist in that same sense.
If by existence you also mean, "something that continues to exist upon closer inspection," then neither particles nor waves exist, you can always design an experiment to refract something around a corner and meausre individual quanta of it when it gets to the detector.
The comment gave a meaning of existence: "...if it can have a transitive causal influence upon us", so there is no need to imagine up straw ones.
The comment defined existence, but didn't define it. The followup commenter was clarifying that proving the is something that exists isn't a proof that a particular claimed thing exists.
"it" is something being referenced - some details in the territory being marked out by the map. The map, and this act, provide no evidence that something exists. The only real basis is if you have evidence there's some details, of the nature claimed by your map, that can have some sort of transitive causal influence upon us.
Zeus and Sherlock Holmes could both be argued to have had transitive causal influences upon us - so in order to make that statement make sense there must be some other implied constraints on what is considered to "exist."
The fictional character Sherlock Holmes certainly exists.
It all depends on what you mean when you say "The fictional character Sherlock Holmes". I believe that thoughts, memories and knowledge "about Sherlock Holmes" exist and are physical details, as are Sherlock Holmes books, etc. And that there's no other reality to "Sherlock Holmes", above and beyond these things.
But really, this discussion is too large to satisfactorily deal with here.
But really, this discussion is too large to satisfactorily deal with here.
> Zeus and Sherlock Holmes could both be argued to have had transitive causal influences upon us
There's some philosophers who'd argue that, but I think the arguments are pretty weak.
There's some philosophers who'd argue that, but I think the arguments are pretty weak.
Then this is easy, we are discussing the concept infinity, hence it is having a causal influence on our actions and so by your definition it exists.
It's not that easy. It comes down to what "the concept of infinity" means. Does Santa Claus exist because the concept of Santa Claus exists? (See my other comments in this subthread for my view of this.)
I would say he does. Santa Clause has a causal influence on the world. The image that we ascribe to 'him' (the concept) is different from the reality of it's influence but that is true for most things..
We characterize all sorts of things in all sorts of ways, trying to model the reality we perceive, but actually those models are also reality and they are not the same as the thing being modelled.
From this standpoint everything fictional exists but whether there is an 'infinite' amount of it is a super annoying question that just keeps going in circles so I don't really care so much what the answer is.
But if it's infinite then there is a god, right? :^)
We characterize all sorts of things in all sorts of ways, trying to model the reality we perceive, but actually those models are also reality and they are not the same as the thing being modelled.
From this standpoint everything fictional exists but whether there is an 'infinite' amount of it is a super annoying question that just keeps going in circles so I don't really care so much what the answer is.
But if it's infinite then there is a god, right? :^)
> From this standpoint everything fictional exists
No they don't. The mental representations exist, the thought processes exist. That doesn't mean the fictional entities exist above and beyond those.
No they don't. The mental representations exist, the thought processes exist. That doesn't mean the fictional entities exist above and beyond those.
What aspect of waves as they exist in our universe is infinite?
If waves are continuous then they are over the real numbers, and there are an infinite number of reals between any two points.
This is specifically addressed in the article - this is a function how our maths represent phenomena:
"Physicists have found it convenient to use the concept as a mathematical idealization where infinity occurs as a limit of large numbers, even though in physical reality there is no infinity of anything. An example is the Fourier series: while mathematically an infinite number of terms are needed to give an exact representation of the function representing an arbitrary waveform, this does not in fact mean that there are infinite physical frequencies occurring in the real physical system."
"Physicists have found it convenient to use the concept as a mathematical idealization where infinity occurs as a limit of large numbers, even though in physical reality there is no infinity of anything. An example is the Fourier series: while mathematically an infinite number of terms are needed to give an exact representation of the function representing an arbitrary waveform, this does not in fact mean that there are infinite physical frequencies occurring in the real physical system."
"There is a duality between zero and infinity, expressed in the elementary identity 1/0 = ∞ . If one side of the duality does not occur in nature, also the other side ought not to."
It wasn't clear to me that the second sentence was sufficiently proved. At first glance it seems reasonable, but the authors went a bit too fast past that claim for me.
It wasn't clear to me that the second sentence was sufficiently proved. At first glance it seems reasonable, but the authors went a bit too fast past that claim for me.
I don't think it's proved at all. "Ought" is an appeal to symmetry. At the same time I don't think that necessarily undermines the argument.
The dual numbers contain infinitesimal numbers but not infinite numbers.
Oppositely, the variable X of a polynomial behaves much like an infinity, but is not invertible.
Their argument depends crucially on whether infinitesimals/infinities are invertible.
Oppositely, the variable X of a polynomial behaves much like an infinity, but is not invertible.
Their argument depends crucially on whether infinitesimals/infinities are invertible.
AFAIK, p-adic quantum mechanics has shown some promise in treating aspects of the "physics of infinity".
It is the application of p-adic analysis to quantum mechanics.
[0] https://en.wikipedia.org/wiki/P-adic_quantum_mechanics
It is the application of p-adic analysis to quantum mechanics.
[0] https://en.wikipedia.org/wiki/P-adic_quantum_mechanics
The article doesn't load for me in Firefox or Chrome. It's stuck at 'Loading enhanced PDF...'.
Doesn't load for me either. Stuck at 'pay us money'.
Statements like this always bring me back to Goedel's incompleteness theorem...
[deleted]
Infinity in nature sounds like an untestable argument. If space was infinite, by definition we'd never be able to measure it's boundary to verify the claim. The closest we could come is "we still haven't discovered a boundary to the volume of space".
Basically the claims "beyond the farthest point you can measure there is more space" and "beyond the farthest point you can measure there is nothing" are equally plausable and untestable.
Basically the claims "beyond the farthest point you can measure there is more space" and "beyond the farthest point you can measure there is nothing" are equally plausable and untestable.
If you had read the article, you'd learn that the problem is much richer and more complicated than the way you stated it.
[deleted]
1/0 is not infinity, it’s undefined.
For the function 1/x as x->0 from the positive side the function tends to positive infinity, while as x->0 from the negative side the function goes to negative infinity.
The function cannot be both plus and minus infinity at the same time.
For the function 1/x as x->0 from the positive side the function tends to positive infinity, while as x->0 from the negative side the function goes to negative infinity.
The function cannot be both plus and minus infinity at the same time.
While you are technically correct, that's pedantic and doesn't impact the conclusions of this article one bit, nor does it really add to the discussion. It's written for the layman, and chose to purposefully ignore technical details such as lim x->0+ 1/x = +inf and lim x->0- 1/x = -inf.
Sure there is an infinity without a sign.
I find the article a bit pointless. The concept of infinity is made up by men. Just like the concept of numbers or logic.
There doesn't exist something like a 'four' in reality, this only exists in models that we have created.
Therefore it goes without saying that there is no 'infinity' in physical reality. On the other hand, nothing is keeping me from creating some model of reality that has infinity in it.
Therefore it goes without saying that there is no 'infinity' in physical reality. On the other hand, nothing is keeping me from creating some model of reality that has infinity in it.
"4" is a concept, but you don't deny that the 4 turnips on your table do have physical reality, do you? Is the notion that there are an infinite number of turnips spread out through an infinite universe "just a concept" whose physical reality we don't have to bother ourselves with, in a way that those 4 turnips aren't?
I will not deny that a collection of atoms, mostly carbon, hydrogen and oxygen, with the rough weight of 4 turnips, is on the table and looks like 4 turnips.
The 4 turnips do have physical reality but on the other hand, the 4 in this scenario is not a realized concept, there is nothing physical about the 4 in this scenario since all the things in it are made of particles made of mostly nothing.
The 4 turnips do have physical reality but on the other hand, the 4 in this scenario is not a realized concept, there is nothing physical about the 4 in this scenario since all the things in it are made of particles made of mostly nothing.
The point in contention isn't that infinities can be imagined, it is whether they are useful - specifically in the context of cosmology.
Numbers are clearly useful, otherwise it would be difficult to explain the importance we ascribe to the value of our bank balances. However just because you can formulate a model of reality including infinities it does not therefore follow that this model must be useful, or that infinities must be useful.
Numbers are clearly useful, otherwise it would be difficult to explain the importance we ascribe to the value of our bank balances. However just because you can formulate a model of reality including infinities it does not therefore follow that this model must be useful, or that infinities must be useful.
"The point in contention isn't that infinities can be imagined, it is whether they are useful - specifically in the context of cosmology."
Going the other way may be more useful; is it possible to create a correct model of reality that does not at any point use infinities? You could have alternate correct models that do, but if you can create one without them you'd have a reasonable basis to call the infinities in the other models unnecessary and nonexistent.
(I would suggest before anyone rush to correct me that they read what I said carefully. For instance, "reasonable basis" != "undeniable proof", which experience leads me to believe is a difference a hypothetical replier might, in their rush to reply, overlook. Also note the two clauses in the question; if it is simply impossible to create a correct model of reality at all, we may never get to the "infinity" portion of the question.)
Going the other way may be more useful; is it possible to create a correct model of reality that does not at any point use infinities? You could have alternate correct models that do, but if you can create one without them you'd have a reasonable basis to call the infinities in the other models unnecessary and nonexistent.
(I would suggest before anyone rush to correct me that they read what I said carefully. For instance, "reasonable basis" != "undeniable proof", which experience leads me to believe is a difference a hypothetical replier might, in their rush to reply, overlook. Also note the two clauses in the question; if it is simply impossible to create a correct model of reality at all, we may never get to the "infinity" portion of the question.)
If you could do that - create a correct model of reality - at all infinities or not, there's a Nobel Prize with your name on it. :)
usefulness is a pragmatic attribute, it does not by itself say that something really exist in reality, unless you believe that everything that is useful exists. I think the OP do not believe that everything that is useful exists in physical reality, therefore seeing four tables does not mean that fours exists in physical reality separated from tables or other objects. So you are probably using different metaphysics..
But sure, since the days of William of Occam pragmatic metaphysics have been popular, but again popularity does not necessary prove something is right. Just explaining both sides of the argument here and trying to resolve the potential missunderstandings
But sure, since the days of William of Occam pragmatic metaphysics have been popular, but again popularity does not necessary prove something is right. Just explaining both sides of the argument here and trying to resolve the potential missunderstandings
> does not mean that fours exists
Of course "they" do - just like anything you care to use a word for, e.g. "humans." All words are, in fact, abstractions, and these abstractions do exist in a meaningful sense.
Of course "they" do - just like anything you care to use a word for, e.g. "humans." All words are, in fact, abstractions, and these abstractions do exist in a meaningful sense.