# What is the size of a point? Or, what is a particle?

#### Naty1

atyy posts:
I think infinities per se are not a problem, but only if they threaten the mathematical coherence of the theory, prevent predictability, or conflict with experimental measurements.
good points.

and

Quantum chromodynamics is a quantum field theory which does not have such troublesome infinities.
is that a bit of a stretch? Or have methods been developed to tame such infinities??

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#### apeiron

Gold Member
That isn't quite right. First order phase transition, by definition, has a discontinuity in a state variable. This clearly implies that the rate of change at such a phase transition is infinite.....
Surely it is just the model which treats the macrostate transition as being infinitely fast, taking no time and being thus discontinuous? In reality, it takes a beat or two for the microstates to correlate, become ordered in a different fashion?

#### ZapperZ

Staff Emeritus
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Surely it is just the model which treats the macrostate transition as being infinitely fast, taking no time and being thus discontinuous? In reality, it takes a beat or two for the microstates to correlate, become ordered in a different fashion?
Er... who said anything about the change being a function of time? You could measure the state variable as a function of anything, such as temperature, volume, etc.

Zz.

#### apeiron

Gold Member
Er... who said anything about the change being a function of time? You could measure the state variable as a function of anything, such as temperature, volume, etc.

Zz.
So the answer is that the infinity is a property of the model, not necessarily a property of the reality.

Which goes to the heart of the OP question. We can model reality in terms of points, but what then is the reality of such points?

#### ZapperZ

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So the answer is that the infinity is a property of the model, not necessarily a property of the reality.
How do you know this?

Zz.

#### seerongo

As the OP, I just wanted to briefly say that I'm glad to see this thread get back on the track that I had hoped it would go from the start. I'm sorry if my intuitive leanings got in the way there for a while, but I must say that I have learned a lot from this discussion. Even though my profession is in a different field, it is almost scary how much I actually understand from you experts, complementing my own readings.

That said, I leave the discussion to you.

PS. I will admit to Zz and all that my favorite color is: 33CCFF...(Please don't reply to this, though, unless you see it as a new thread in General Discussion which I see is a bit more relaxed).

Thanks to all.

Ron

#### apeiron

Gold Member
How do you know this?

Zz.
What? How do I know it is a property of a model? Or how do I know it is not necessarily a property of reality?

The answer generally would be that I believe the human mind to be in a modelling relationship with reality and these are the consequences of this view.

Do you take some different position here? Naive realism? Platonism?

#### ZapperZ

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2018 Award
What? How do I know it is a property of a model? Or how do I know it is not necessarily a property of reality?

The answer generally would be that I believe the human mind to be in a modelling relationship with reality and these are the consequences of this view.

Do you take some different position here? Naive realism? Platonism?
None. I'm an experimentalist, and I accept things as valid when there are valid empirical evidence to support them.

So you accept as a fact that there is a difference between our description of the world, and the world itself? That's why I asked how you would know this. As far as I can tell, your argument so far as been based on nothing but a matter of tastes.

Zz.

#### apeiron

Gold Member
None. I'm an experimentalist, and I accept things as valid when there are valid empirical evidence to support them.
This is naive and unrigorous.

What are the "things" that are validated by experiments if not models?

Take a concept like temperature for example. In what way is this not a model? Or even a family of models (from caloric fluids to boltzmann ensembles)? And where is the "reality" in our sensations of hot and cold?

Empiricism means you test your ideas against observation. But you have to have crisp and formal ideas (models, theories, hypotheses) to know what kind of observations would count as a valid test.

#### Eelco

Interesting replies. The above quote exactly summarizes what has always been difficult for me. I think I just have a conceptual problem with the idea of a continuum of space and time). For some reason, the idea of a quantum space/time actually makes more intuitive sense to me than a continuum. I don't know why because both are hard to imagine, but it has to be one or the other in the absence of alternatives (are there any?). A continuum seems to lead inevitably to physical reality or existence of infinite and infinitesimal objects, space, time, ad infinitum (no pun intended) and that is what I can't get past. Naty1's reference to T dualities is another example of this problem. No infinities for me, please.

Ron

Non-finitist mathematics is strictly speaking, not logical. It is riddeled with internal contradictions, and thus on the most elementary logical grounds, should have to be rejected.

Perhaps you are aware of Gallileo's paradox, the answer to the question 'are there less squares of natural numbers than there are natural numbers', being both yes and no. A contradiction, plain and simple.

Contemporary mathematicians 'resolve' this contradiction by saying that an infinite set is one that can be put into one-to-one correspondance with a subset of itself. Its a negation of identity, of course, but if you state it without flinching your eyes, most mathematicians will buy it and consider the issue resolved.

Infinite divisibility isnt any better. If you are feeling bored, show some mathematicians how to put the real numbers in one to one correspondence with the natural numbers (the proof is not particularly hard to find). Its even better than arguing with creationists. Mathematicians would rather bite off their tongues than admit to internal contradictions in their trade.

Whether or not reality is logical is a metaphysical belief that I cant really argue one way or another, but I have yet to see the first instance of reality contradicting itself, and I think the infinite divisibility of space will go the same way as the infinite extent of the earth, the infinite extent of the universe, and the infinite divisibility of matter.

Choosing between non-contradiction and lorentz invariance seems like an easy choice to me.

#### Eelco

That isn't quite right. First order phase transition, by definition, has a discontinuity in a state variable. This clearly implies that the rate of change at such a phase transition is infinite.

The van Hove singularity is very common in your semiconductor.

The density of states of a superconductor right at the energy gap edge is infinite.

The spectral function of a quasiparticle in an ordinary metal, which is the imaginary part of the single-particle Green's function, is essentially a delta function. This gets you the ordinary metallic behavior that gives you all the Drude model that we know and love.

Etc... etc.
These are all approximate models of reality. There are no first order phases transitions, there are no semiconductors of infinite extent, there are no quasiparticles.

These infinities arise for no other reason than you (implicitly) demanding they be in your model.

We have no issues with dealing with an infinite value. A whole branch of mathematics is built around dealing with poles like this.
Mathematics deserves some qualification. If you meant logically inconsistent mathematics, ok.

#### ZapperZ

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These are all approximate models of reality. There are no first order phases transitions, there are no semiconductors of infinite extent, there are no quasiparticles.

These infinities arise for no other reason than you (implicitly) demanding they be in your model.
These so-called "models" work, and you depend your life into it.

And who demanded a semiconductor of infinite extent?

I could also say that your assertion is also an "approximate model of reality", assuming we all know what "reality" really is.

Let's put it this way, anytime someone tells me that there is a difference between our physical description of the world we live in and "reality", I would appreciate an evidence to show me this difference. Till you can show that, save your breath and your effort.

Zz.

#### ZapperZ

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2018 Award
This is naive and unrigorous.

What are the "things" that are validated by experiments if not models?

Take a concept like temperature for example. In what way is this not a model? Or even a family of models (from caloric fluids to boltzmann ensembles)? And where is the "reality" in our sensations of hot and cold?

Empiricism means you test your ideas against observation. But you have to have crisp and formal ideas (models, theories, hypotheses) to know what kind of observations would count as a valid test.
But this is exactly what I'm asking. How are you to know that what you are saying here is valid. You make it seem as if there is a CLEAR difference between what we describe, and what "reality" is. For all I know, your assertion that these are "models" and not "reality" is in itself, A MODEL! Not only that, it is a model of the worst kind - it has no empirical basis.

Zz.

#### Eelco

These so-called "models" work, and you depend your life into it.

And who demanded a semiconductor of infinite extent?

I could also say that your assertion is also an "approximate model of reality", assuming we all know what "reality" really is.

Let's put it this way, anytime someone tells me that there is a difference between our physical description of the world we live in and "reality", I would appreciate an evidence to show me this difference. Till you can show that, save your breath and your effort.
Ehm, dont take my word for it, but the way i understand it, the atomic hypothesis has a fair amount of credibility these days. A continuum of state variables is recognized as an approximation. Inviscid compressible flow equations develop infinite gradients. What of it?

The relevant question is whether our most reductionistic theories of space, time and matter require a notion of infinity. It is not at all obvious that they do.

#### ZapperZ

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Ehm, dont take my word for it, but the way i understand it, the atomic hypothesis has a fair amount of credibility these days. A continuum of state variables is recognized as an approximation. Inviscid compressible flow equations develop infinite gradients. What of it?

The relevant question is whether our most reductionistic theories of space, time and matter require a notion of infinity. It is not at all obvious that they do.
I'm not sure the relevance of what you just wrote here. And as far as "reductionistic theories of space, time, and matter" is concerned, if you do a search of "emergent phenomena" on here. You'll find many of my posts that try to go against such reductionist approach. So I'm not the one you should be arguing against.

Zz.

#### Eelco

I'm not sure the relevance of what you just wrote here.
The relevance is that I personally do expect fundamental physics to be logically consistent.

First order phase transitions are not fundamental physics. It is an approximate model to what we do believe to be closer to the truth, at least. Invoking the mathematical needs of such approximate models is in my opinion not at all an argument in favor of infinity.

And as far as "reductionistic theories of space, time, and matter" is concerned, if you do a search of "emergent phenomena" on here. You'll find many of my posts that try to go against such reductionist approach. So I'm not the one you should be arguing against.
I too think a lot of reductionism is misguided, but im all for reductionism, insofar as this means trying to find a minimum set of principles which explain all physics. No clue what you are referring to here, so perhaps you are doubly the one i should be arguing against.

#### ZapperZ

Staff Emeritus
2018 Award
The relevance is that I personally do expect fundamental physics to be logically consistent.

First order phase transitions are not fundamental physics. It is an approximate model to what we do believe to be closer to the truth, at least. Invoking the mathematical needs of such approximate models is in my opinion not at all an argument in favor of infinity.
I don't invoke the needs. I described the current understanding of not only the classical phase transition, but also quantum phase transition.

Again, this "truth" and this "reality" needs to established. You again are arguing there such a thing exist without any justification.

Zz.

#### Eelco

I don't invoke the needs. I described the current understanding of not only the classical phase transition, but also quantum phase transition.

Again, this "truth" and this "reality" needs to established. You again are arguing there such a thing exist without any justification.

Zz.
Reality is an assumption I hold on to, yes. If you dont, good for you, but then this discussion is over.

#### ZapperZ

Staff Emeritus
2018 Award
Reality is an assumption I hold on to, yes. If you dont, good for you, but then this discussion is over.
It would be nice if you know when you actually have found this "reality".

This was the point I was trying to make, which no one could point to. So essentially this discussion was over a long time ago.

Zz.

#### Eelco

It would be nice if you know when you actually have found this "reality".
I define it as 'that which I can see'. That makes it easy, as you dont have to bother to expend any energy going looking for it.

Models should fit the data, and they should not contradict themselves. Within those constraints, anything goes as far as im concerned.

#### ZapperZ

Staff Emeritus
2018 Award
I define it as 'that which I can see'. That makes it easy, as you dont have to bother to expend any energy going looking for it.
Then you need to argue with apeiron, who said:

apeiron said:
Empiricism means you test your ideas against observation. But you have to have crisp and formal ideas (models, theories, hypotheses) to know what kind of observations would count as a valid test.
So your observation is based on at least a theoretical description/model, whether you are aware of it or not.

Zz.

#### apeiron

Gold Member
But this is exactly what I'm asking. How are you to know that what you are saying here is valid. You make it seem as if there is a CLEAR difference between what we describe, and what "reality" is. For all I know, your assertion that these are "models" and not "reality" is in itself, A MODEL! Not only that, it is a model of the worst kind - it has no empirical basis.

Zz.
What kind of sophomoric epistemological point are you trying to make here? That we must doubt, and then doubt even our doubts?

Of course I am talking about the model of the model. The meta-model. And being on different epistemic levels, doubt does not apply in exactly the same way. Though of course doubt still must apply.

And of course the meta-model has an empirical basis - a general one appropriate to its level rather than a particular one that would be appropriate to some particular model (or theory).

So, to recapitulate the approach we have been taking since the Enlightenment, we realise that the human mind does not know the world directly. Ideas shape our impressions (and it being a modelling relationship, impressions in turn are shaping our ideas).

I note you ducked the question about temperature. But of course you knew that this was one of the most ancient sources of doubt, the way that if you stick one hand in hot water, the other in cold, then put both hands into lukewarm water and get conflicting "empirical" reports.

So given that our ideas and impressions need to be developed in some more rigorous fashion, we invented the scientific method. We created formal models (equations, that kind of stuff) which made predictions that we could test via measurements, experiments, and then changed the equations in the light of what we observed or learnt.

And we also gave away the idea that the equations needed to be representations of "how things really are".

Remember Newton and his famous struggles to believe in action at a distance. Hypotheses non fingo and all that. This part of his theory of gravitation seemed unreal to everyone and it most probably is. But as a model, the stripped down and unreal view is usually what works.

Of course, some fools then believe that the model IS then the real. No. What we have done is accept we are modellers and so therefore all our views of reality are to some extent compromised, unreal, fictional - but in a useful way.

Newton's gravity was supported by certain lines of empirical evidence. And his general approach to modelling - where the unreality of the equations ceases to be an issue, just part of the game being played - has been supported empirically by a lot of successful science since.

The meta-modelling is empirically supported by the success of the modelling!

So nothing I say here is a matter of taste. It is simply a modern 21st Century way of more precisely stating the direction we have been going since at least the Enlightenment.

Believe it or not, the epistemology of science (and understanding generally) is still a work in progress. There are new details to add. There are more rigorous meta-models being developed.

So unless you have studied the recent literature - and all your reponses seem strictly personal, off the top of your head impressions, lacking any grounding in citations - you are going to struggle to make a telling point here.

#### ZapperZ

Staff Emeritus
2018 Award
What kind of sophomoric epistemological point are you trying to make here? That we must doubt, and then doubt even our doubts?

Of course I am talking about the model of the model. The meta-model. And being on different epistemic levels, doubt does not apply in exactly the same way. Though of course doubt still must apply.
So then what makes your model any more "real" than mine?

To address your "temperature" model, first of all this has nothing to do with what we consider as "hot or cold". This is a subjective sensation and has nothing to do with physics. Secondly, there's a difference between (i) a valid model that later on gets enclosed into a more general description versus (ii) a flawed model. Newton's laws are examples of (i) and the caloric/Bohr atom are the 2nd. It does mean that I consider Newton's laws as a valid description of "reality" within the realm of their applicability.

So, to recapitulate the approach we have been taking since the Enlightenment, we realise that the human mind does not know the world directly. Ideas shape our impressions (and it being a modelling relationship, impressions in turn are shaping our ideas).
But again, how do you know this? "does not know the world directly" is a loaded statement. It says nothing about what it means as "knowing" and it says nothing about what it means to know something "directly", as if there is such a thing. Did you directly read all these things that I wrote?

So nothing I say here is a matter of taste. It is simply a modern 21st Century way of more precisely stating the direction we have been going since at least the Enlightenment.
No, I still say it is a matter of tastes. I can easily point to the "shut-up-and-calculate" philosophy in which even demanding that there is a difference between what we perceive and what reality is is a meaningless dichotomy, because you have no empirical evidence that there is such a difference. Even extreme solipsism would argue alongside with that. I only need to point to only one different point of view here to show other possible point of view to show that adopting one IS a matter of preference.

Zz.

#### apeiron

Gold Member
But again, how do you know this? "does not know the world directly" is a loaded statement. It says nothing about what it means as "knowing" and it says nothing about what it means to know something "directly", as if there is such a thing. Did you directly read all these things that I wrote?

.
Sorry, you'll have to come out of the closet here and tell us what it is you actually do believe.

Are you saying we DO directly know the world in unmediated fashion?

Are you saying ALL epistemologies are a matter of taste?

What are the empirical grounds that support your chosen epistemology?

No, I still say it is a matter of tastes. I can easily point to the "shut-up-and-calculate" philosophy in which even demanding that there is a difference between what we perceive and what reality is is a meaningless dichotomy,
.

Your own arguments presented here are simply naive and incoherent.

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