Living Opponents of the Copenhagen Interpretation

[bhobba]

Yes, I am aware that QM has a measurement problem. But historically the measurement problem combined with the positivist idea that "only the measurable is meaningful" to the actual belief that the unmeasurable is nonexistent, which a stop to inquiry, as the Weinberg article above spells out

You are getting a bit confused between a theory that is silent about things other than observations, and philosophical guff that says that's all there is.

Thanks
Bill

 

[Fredrik]

That's a misconception.

Again the statement you're disagreeing with is something that I would say is definitely true. There is no piece of mathematics that says anything about the real world all on its own. Hilbert space theory doesn't say anything about the result of experiments. To turn it into a theory of physics, we have to add correspondence rules that tell us how to interpret the mathematics as predictions about possible results of experiments. The same goes for every piece of mathematics and the corresponding application to the real world. In the theory I defined in post #86, everything was pure mathematics that said nothing about the real world until I made that final assumption, the correspondence rule.

In some cases, it's so obvious what the correspondence rules are supposed to be that we may not even realize that we are using correspondence rules. Euclidean geometry is probably the best example. It's a piece of pure mathematics that was intended to be applied to the real world. We have known since we were kids how mathematical things in Euclidean geometry correspond to real-world things. Those correspondences are what turn this piece of pure mathematics into a theory of applied mathematics.

In general, I would say that the difference between pure mathematics and applied mathematics is correspondence rules. Applied mathematics is a subject that includes applied geometry, applied probability theory and physics.

 

[bhobba]

Kant's idea that time and space are mind-dependent seems to me to be a philosophical precursor to relativity.

Its the other way around.

Kant's carry on about the a priori nature of Euclidean geometry negatively affected the equally great mathematician Gauss from publishing his discoveries about non Euclidean geometry. This was required for Riemann to develop Riemannian geometry, the extension of that to pseudo Riemannian geometry being the basis of General Relativity.

Thanks
Bill

 

[jbmolineux]

Weinberg is dismissing the idea that philosophers can tell physicists which theories are plausible and which ones are not. I think everyone in physics agrees with him. But the idea that you can't find the right theory just by thinking about it is a philosophical principle that it's important to get right. So it would be more accurate to say that Weinberg is supporting the quoted sentence than to say that he's disagreeing with it.Now this is something I strongly disagree with. bhobba's example of Euclidean geometry shows that a theory can make good predictions even if's only an approximate description of the world. Here's an example of a theory that makes good predictions without even approximately describing the world:

Define $\Omega=\{1,2,3,4,5,6\}$. Let $\Sigma$ be the set of all subsets of $\Omega$. For each $E\in\Sigma$, let $|E|$ denote the cardinality of $E$, i.e. the number of distinct elements of $E$. Define $P:\Sigma\to[0,1]$ by
$$P(E)=\frac{|E|}{|\Omega|}$$ for all $E\in\Sigma$. Now let's turn this piece of mathematics into a theory about the real world by specifying that for each $E\in\Sigma$, $P(E)$ is the fraction of times we'll get a result in the set $E$ if we repeatedly throw a standard six-sided die a large number of times.

This theory isn't even approximate description of the world. In particular, it says nothing about what's actually happening to the die between state preparation (the throw) and measurement (the moment when it has landed with some side up).

Weinberg is dismissing the idea that philosophers can tell physicists which theories are plausible and which ones are not. I think everyone in physics agrees with him. But the idea that you can't find the right theory just by thinking about it is a philosophical principle that it's important to get right. So it would be more accurate to say that Weinberg is supporting the quoted sentence than to say that he's disagreeing with it.Now this is something I strongly disagree with. bhobba's example of Euclidean geometry shows that a theory can make good predictions even if's only an approximate description of the world. Here's an example of a theory that makes good predictions without even approximately describing the world:

Define $\Omega=\{1,2,3,4,5,6\}$. Let $\Sigma$ be the set of all subsets of $\Omega$. For each $E\in\Sigma$, let $|E|$ denote the cardinality of $E$, i.e. the number of distinct elements of $E$. Define $P:\Sigma\to[0,1]$ by
$$P(E)=\frac{|E|}{|\Omega|}$$ for all $E\in\Sigma$. Now let's turn this piece of mathematics into a theory about the real world by specifying that for each $E\in\Sigma$, $P(E)$ is the fraction of times we'll get a result in the set $E$ if we repeatedly throw a standard six-sided die a large number of times.

This theory isn't even approximate description of the world. In particular, it says nothing about what's actually happening to the die between state preparation (the throw) and measurement (the moment when it has landed with some side up).

Fredrik, I believe that theory IS describing the world. It's describing the fact that a six-side die is a cube which--if it is spun into the air and allowed to hit a hard surface from an undetermined height, at an undetermined angle and rate of rotation--are each equally-likely to end up facing upwards when it stops bouncing.

Obviously if you take out the "undetermined height, angle, and rate of rotation" aspects, the theory would not hold. I think Carl Popper describes it as a "propensity" that exists in the die which, by the very nature of its shape, leads to these results when it is repeatedly dropped in the way described above.

Or does anyone have another explanation of why the theory actually holds? Does it really have nothing to do with the shape of the die? If it doesn't have to do with the shape of the die (as described above), why does it work with a normal six sided die, and not a weighted die?--or an 8-side die?

Heck, I've never even tested that theory in a laboratory and I know it holds just by knowing the shape of the die! Has anyone here tested it in a laboratory? Since physics is just about the results of experiments about what can be repeatedly testable, wouldn't we have to test it in a lab to even be able to say that it holds?--and yet, having neither tested it in a laboratory nor bothered to read about what-we-already-know-would-happen if someone DID test it, we all know that the theory is true!

 

[bhobba]

Again the statement you're disagreeing with is something that I would say is definitely true.

I usually agree with your views.

However here I must dissent.

Euclidean Geometry as usually presented is a mathematical model whose statements can be tested eg do the angles of a triangle add up to 180%.

Hilbert's treatment however is another matter:
http://en.wikipedia.org/wiki/Hilbert's_axioms

Hilbert space theory doesn't say anything about the result of experiments.

No it doesn't. Its the mapping of resolutions of the identity to the primitive of observations that does that. That's what QM does - and that's what makes it a mathematical model.

The usual presentation of Euclidean geometry speaks of points of no size and lines of no width that are idealisations of stuff out there.

Thanks
Bill

 

[jbmolineux]

You are getting a bit confused between a theory that is silent about things other than observations, and philosophical guff that says that's all there is.

Thanks
Bill

Bill, if scientific theories were actually "silent about things other than observations" then they wouldn't be able to report anything other than those observations themselves. It's precisely their ability to say MORE THAN the observations themselves that makes them valuable! I certainly agree that its "philosophical guff" which says "that's all there is" but I think its a mistake to say that reporting-only-the-content of observation is all that the theory is saying!

 

[atyy]

The only reason for this is that what most people think of as QM is a great theory of physics plus the unnecessary and unscientific assumption that a pure state "provides a complete and exhaustive description of an individual system". Quantum mechanics as I would define it, doesn't have a measurement problem. There are things that this theory is unable to tell us, but that's not a problem.

I basically agree with you for the definition of "complete" you are using here: that a more complete theory than quantum mechanics cannot exist. In other words, you are saying that quantum mechanics is incomplete, like our other theories including Newtonian gravity, classical special relativity, classical general relativity, non-relativistic quantum mechanics, and the standard model of particle physics. However, where I slightly disagree is that I think a full solution to the measurement problem should also provide explicit examples of what the more complete theory is.

There are two classes of theory. The first class is potentially complete as a model of reality. This class requires experiments to tell us that the theory is incomplete. As an example, I would say give the classical Maxwell's equations without point charges. It is conceivably a theory of some reality independent of observers. However, experiments tell us that the classical Maxwell's equations are incomplete - there is also gravity, and there is quantum mechanics.

The second class of theory makes successful predictions, but we can know that they are incomplete even before they are falsified. As an example, I would give the standard model of particle physics, or quantum Einstein gravity. Both theories match almost all observations (except the neutrino mass and dark matter), yet we know that they are incomplete, because they fail to be predictive or even consistent above a certain energy. It is for this reason that there are scientific research programmes asking what the completion of quantum gravity is. Similarly, Bohmian mechanics, Many-Worlds, constraints on hidden variables by Bell are scientific research programmes asking what the possible completions of quantum mechanics are.

As an aside, I do realize you are using Ballentine's words to describe what he is opposing. Let's switch from quantum mechanics to Ballentine interpretation. "A pure state is a complete and exhaustive description of an individual system" is correct quantum mechanics, since it just means complete within the theory, which is a choice of system/apparatus, operators, commutation relations, Hilbert space. Similarly, when I assert that "momentum and position are the complete and exhaustive description of an individual system" in single particle Newtonian mechanics, I am not asserting that Newtonian mechanics is the final theory.

Edit: I think I might disagree with you in a major way - at a quick reading, Wikipedia's description of the measurement problem is incorrect.

 

[atyy]

Or maybe I just don't understand the Bell Test experiments, and how they somehow vindicate some philosophical ideas that I have been considering errors.

I read the rest of your post too, but I'll need to think a bit before a more detailed reply. But here I just want to say that it isn't right to claim that the Bell Test experiments vindicate "nonrealism" (whatever that means). Rather the Bell Test experiments say that if we believe in reality, then it is nonlocal in a certain sense. In particular, the predictions of quantum mechanics cannot be generated by any local deterministic theory. I should also say the experiments at present do have loopholes, so we are not forced to accept that reality is nonlocal. Also, even if the experiments were perfect, there are (at least) two loopholes that would allow reality to be local. The first loophole is superdeterminism, and the second loophole is retrocausation.

 

[carllooper]

Carl, I certainly don't believe that any philosophical error on Einstein's part caused any problems in physics! Kant's idea that time and space are mind-dependent seems to me to be a philosophical precursor to relativity.

Yes, different people have different ideas of what constitutes "reality," "non-existent," "particle," etc.--as you point out. But then you go on to say that "as long as one understands what is being meant"--that’s what matters. But since there is no agreement about what is meant by those terms, how can it be understood what is being meant?

Further, the question of whether something even exists if it can't be measured by the technology of a certain time is certainly relevant! If scientists are taught something doesn't exist, it puts a stop to the inquiry that drives scientific progress!

I didn't mention anything at all about mind-dependence in Kant.

Regarding understanding. Let's suppose you are reading a paper in which someone claims the wave function is real. What you really want to know by that is what they mean by that - what do they mean by "real". It seems such a simple thing but it's not. There's a history there. And you need to work through that and the context in which the paper is presented, to get an idea of what they mean by that word "real". Once you understand what they are on about it almost doesn't matter whether they had used the word "real" or said something like the "wave function is a chicken".

C

 

[jbmolineux]

I read the rest of your post too, but I'll need to think a bit before a more detailed reply. But here I just want to say that it isn't right to claim that the Bell Test experiments vindicate "nonrealism" (whatever that means). Rather the Bell Test experiments say that if we believe in reality, then it is nonlocal in a certain sense. In particular, the predictions of quantum mechanics cannot be generated by any local deterministic theory. I should also say the experiments at present do have loopholes, so we are not forced to accept that reality is nonlocal. Also, even if the experiments were perfect, there are (at least) two loopholes that would allow reality to be local. The first loophole is superdeterminism, and the second loophole is retrocausation.

Yes, that was my understanding as well--although I've heard it both claimed that the Bell Test experiments vindicated "non-realism" and that it was simply LOCAL realism that they violated. But that leads me to another question--why is non-local realism so hard to fathom? What's so spooky about "action at a distance?" Didn't that die with gravity and electromagnetism? Don't those both clearly involve action-at-a-distance? (Or is it that they are not instantaneous but travel at the speed of light?)

 

[jbmolineux]

I didn't mention anything at all about mind-dependence in Kant.

Regarding understanding. Let's suppose you are reading a paper in which someone claims the wave function is real. What you really want to know by that is what they mean by that - what do they mean by "real". It seems such a simple thing but it's not. There's a history there. And you need to work through that and the context in which the paper is presented, to get an idea of what they mean by that word "real". Once you understand what they are on about it almost doesn't matter whether they had used the word "real" or said something like the "wave function is a chicken".

C

Yeah, but the "history" there is a mess!

 

[bhobba]

Bill, if scientific theories were actually "silent about things other than observations" then they wouldn't be able to report anything other than those observations themselves

Scientific theories in general aren't silent about such things - only some like QM - probability theory is another example.

But again what you say is a misconception - QM contains the full machinery of Hilbert space for example, and that contains all sorts of things. Its only specific things that are mapped to stuff out there. And indeed it is those other things that often lead to powerful consequences such as Gleason's Theorem and the existence of Born's rule.

Thanks
Bill

 

[bhobba]

why is non-local realism so hard to fathom? What's so spooky about "action at a distance?" Didn't that die with gravity and electromagnetism? Don't those both clearly involve action-at-a-distance? (Or is it that they are not instantaneous but travel at the speed of light?)

Non local realism is a-priori just as valid as its converse.

A choice, without direct experimental support, reveals more about the prejudices of the person concerned than 'reality' - just like Einstein's belief QM was incomplete - and Bohr's belief it was. Neither are right - or wrong.

Personally I think QM violates both locality and realism of naive realism - but that's just me and simply tells you about my world view - it means diddly squat.

That said it doesn't remove the responsibility of actually having an opinion.

Thanks
Bill

 

[carllooper]

Yeah, but the "history" there is a mess!

So? Do you think the history of philosophy is any tidier? Or the history of any other discipline? There's no magic wand to be found and waved and everything suddenly becomes tidy.

 

[bhobba]

So? Do you think the history of philosophy is any tidier? Or the history of any other discipline? There's no magic wand to be found and waved and everything suddenly becomes tidy.

Especially philosophy. Do you know of a single issue (other than something trite) that philosophers agree on? Mathematicians and physicists agree on all sorts of things. And in general they make progress - not philosophers:
http://www.ralphmag.org/EQ/gauss-kant.html

Rather amusing - but still illustrating an important point.

Thanks
Bill

 

[Fredrik]

Fredrik, I believe that theory IS describing the world. It's describing the fact that a six-side die is a cube which--if it is spun into the air and allowed to hit a hard surface from an undetermined height, at an undetermined angle and rate of rotation--are each equally-likely to end up facing upwards when it stops bouncing.

Obviously if you take out the "undetermined height, angle, and rate of rotation" aspects, the theory would not hold. I think Carl Popper describes it as a "propensity" that exists in the die which, by the very nature of its shape, leads to these results when it is repeatedly dropped in the way described above.

Or does anyone have another explanation of why the theory actually holds? Does it really have nothing to do with the shape of the die? If it doesn't have to do with the shape of the die (as described above), why does it work with a normal six sided die, and not a weighted die?--or an 8-side die?

To say that a theory describes (an aspect of) the real world should mean that the purely mathematical part of the theory describes a fictional universe that bears a strong resemblance to (that aspect of) the real world. In this case, the purely mathematical part of the theory doesn't contain concepts such as "shape", "die", "velocity", "angular momentum", etc., so I would say that it definitely doesn't describe those aspects of the real world.

You asked for an explanation of why the theory holds. The only thing that can explain why a theory holds is a better theory, and there are undoubtedly many theories that are better than this one in the sense that they can reproduce all the predictions of my theory, and make a few new ones. One such theory is classical mechanics, but I'm sure that if we tried, we could come up with something a lot worse than that, but still better than my theory, that includes concepts like "shape". The fact that it's intuitively obvious to you that such a theory must exist doesn't mean that my simple theory describes what's going on in terms of things like "shape".

Heck, I've never even tested that theory in a laboratory and I know it holds just by knowing the shape of the die! Has anyone here tested it in a laboratory? Since physics is just about the results of experiments about what can be repeatedly testable, wouldn't we have to test it in a lab to even be able to say that it holds?--and yet, having neither tested it in a laboratory nor bothered to read about what-we-already-know-would-happen if someone DID test it, we all know that the theory is true!

We've all rolled enough dice to be fairly certain what would happen, but yes, the shape of a standard die is what makes it seem obvious to us that the theory must be very accurate. We will think about the theory in terms of "shape" and other familiar concepts from better theories, but the theory itself doesn't contain them.

 

[DennisN]

Further, the question of whether something even exists if it can't be measured by the technology of a certain time is certainly relevant! If scientists are taught something doesn't exist, it puts a stop to the inquiry that drives scientific progress!

Hi, jbmolineux! I just want to mention this short text by Anton Zeilinger:, which was a response to a general question "2014 : What Scientific Idea Is Ready For Retirement?"

The idea to be abandoned is the idea that there is no reality in the quantum world.
[...]
But, whether it has a well-defined position or not, the buckyball very well exists. It is real in the double-slit experiment, even when it is impossible to assign its position a well-defined value.

 

[carllooper]

Yes, that was my understanding as well--although I've heard it both claimed that the Bell Test experiments vindicated "non-realism" and that it was simply LOCAL realism that they violated. But that leads me to another question--why is non-local realism so hard to fathom? What's so spooky about "action at a distance?" Didn't that die with gravity and electromagnetism? Don't those both clearly involve action-at-a-distance? (Or is it that they are not instantaneous but travel at the speed of light?)

The spookiness of action-at-a-distance has it's origins in the fact that action-at-a-distance was once considered, by certain ruling powers, to be supernatural. In and of itself this wouldn't be a problem. After all it's just a word. But anyone found studying the supernatural could be punished. And that's where the word acquires meaning and a certain influence on a culture. Scientists, however, were able to side-step the ban on studying the supernatural by calling what they studied "natural' instead. Perhaps gravity might been formalised earlier had not this taboo around action-at-a-distance been so heavily suppressed when it was.

Or maybe there's some other reasons for the reservations.

In any case it's sometime easier just to retain the language in which a concept has been historically framed. So we could, for example, call non-local realism: 'spooky', but without in any way requiring that non-local realists be sentenced to hang at dawn. In the same way we can retain the word "planet" without in any way suggesting that the so called continue to be considered as wandering gods.

C

 

[atyy]

Yes, I am aware that QM has a measurement problem. But historically the measurement problem combined with the positivist idea that "only the measurable is meaningful" to the actual belief that the unmeasurable is nonexistent, which a stop to inquiry, as the Weinberg article above spells out. I have heard that later the repudiation of "completeness" by Bell came to be accepted, but obviously decades of research-that-could've-been were lost by what were ultimately philosophical mistakes.

To be fair to the early quantum mechanicians, they did realize the problem and discussed it extensively. That is why we still have language like the "Heisenberg cut" and "von Neumann chain". Von Neumann's model of measurement is still one way of presenting the measurement problem that the reversible time evolution of the wave function alone does not produce definite experimental outcomes, contrary to experience. Von Neumann did make an error, but he could hardly have made the wrong claim if he had not been looking for a solution. Also, we know that not everyone accepted von Neumann's erroneous proof before Bell. Bohm, for example, produced a hidden variables theory that reproduces non-relativistic quantum mechanics. Also, Messiah's famous textbook does state that hidden variables that Einstein advocated was not excluded, and that he would adopt Copenhagen just because it was consistent with all experiments up to then.

The most common modern flavour of the Copenhagen interpretation does not deny a common-sense reality, but in fact explicitly states its existence, together with the idea that the wave function is a tool to describe that reality. I will follow the spirit of modern Copenhagen-style interpretations, such as Leifer and Spekkens, who say "the picture we have in mind is of the quantum state for a region representing beliefs about the physical state of the region, even though we do not yet have a model to propose for the underlying physical states."

• Why is it impossible to know both the position and the velocity of a particle by the rebound of a photon hitting it?

It is impossible to know both the position and momentum of a quantum particle, because it does not and cannot have both position and momentum simultaneously. Quantum position and and quantum momentum refer to the results of different experimental procedures, and the procedures are such that you cannot perform both of them in the same place at the same time. We consider these procedures to measure "position" and "momentum", because the quantum "properties" reduce to the classical properties in the classical limit of quantum mechanics.

• On Entanglement experiments - why can't the cause be explained by the same properties in each at the source? (I know I am out of my league with this question, and I believe the answer might be Bell's theorem itself...is that true?)
• Why couldn't there be an entanglement-type experiment where both particles were sent out "in the same way" so that you learn the information from one to know about the other? In other words, can you do something at the beginning to ensure that the position and velocity are the same, and then measure the position of one and the velocity of the other?

That is Bell's great achievement to show that if quantum mechanics is correct, the underlying reality cannot be explained by local properties, for some reasonable definition of "local".

Yes, that was my understanding as well--although I've heard it both claimed that the Bell Test experiments vindicated "non-realism" and that it was simply LOCAL realism that they violated. But that leads me to another question--why is non-local realism so hard to fathom? What's so spooky about "action at a distance?" Didn't that die with gravity and electromagnetism? Don't those both clearly involve action-at-a-distance? (Or is it that they are not instantaneous but travel at the speed of light?)

It is considered "spooky" because Einstein liked to use colourful language. I'm not sure what Einstein had in mind, but it seems to be that spooky action of a distance is in conflict with a classical conception of special relativity, in which the action at a distance is instantaneous and faster than light.

At present, we know that quantum spooky action at a distance is not compatible with any local deterministic theory, but it does not allow faster than light communication, and is compatible with special relativity.

Are there models for the underlying reality compatible with quantum mechanics and relativity? That is still a matter of research. My personal view is that one promising area is to assume that relativity is not exact, for example in lattice models of quantum field theory. However, a problem with that approach is that at present we do not have a lattice model of chiral fermions interacting with nonabelian gauge bosons.

Two very good reviews of the measurement problem are (each is perhaps slightly biased in different ways, but I believe they have striven to be technically accurate):

http://arxiv.org/abs/quant-ph/0209123
Do we really understand quantum mechanics?
Franck Laloe

http://arxiv.org/abs/0712.0149
The Quantum Measurement Problem: State of Play
David Wallace

I should also mention the Transactional Interpretation, which I don't know enough about to know if it is technically right, but is interesting because it tries to use the retrocausation loophole in the Bell Tests.

 

[jbmolineux]

Especially philosophy. Do you know of a single issue (other than something trite) that philosophers agree on? Mathematicians and physicists agree on all sorts of things. And in general they make progress - not philosophers:

I agree completely. The mess that exists currently in philosophy is significantly worse than that of the sciences. Moreover, as the Weinberg article suggested, it is actually RESPONSIBLE for much of the mess in the sciences--which is essentially my major thesis. There are a handful of things about which most philosophers today generally agree, however, and among them is that the philosophy that underlay the development of QM is self-refuting. But my minor thesis is that it still continues to wreak havoc in the sciences today through the idea that "the measurement is all that is being measured" (or "shut up and calculate" or however you want to say it), which is just an extension of that self-refuting philosophical error that has now been discredited in the field of philosophy, but continues to dominate the philosophy of quantum physicists.

Thus, as Atyy says, "It is impossible to know both the position and momentum of a quantum particle, because it does not and cannot have both position and momentum simultaneously." But originally, for the founders of QM, that was just an extension of the empirical criteria of meaning to the theoretical limit of the measurements of that day (and, as I understand it, was directly related to the technological limits of spectroscopes of that time).

But it has become part of the orthodoxy of QM, and physicists today believe themselves to be simply eschewing philosophical bias ("shutting up and calculating") in accepting that. But it seems to me that they are not only inheriting a bias, but they are swallowing whole a philosophical error that is now discredited for being self-refuting, which both historically and logically is the direct and only precursor to the idea that the particles literally "do not have both a position and a velocity."

By the way, I am not suggesting that physics needs to go the field of academic philosophy in order to learn. They need less, not more, of the filth that has been coming out of that decadent institution for the last century or more. By "good philosophy" I just mean careful, accurate, wise thinking about the principles that underlie science, and on questions such as the ones that we're discussing, and those that have become deeply intertwined in physics and particularly QM.

 

[Nugatory]

Thus, as Atyy says, "It is impossible to know both the position and momentum of a quantum particle, because it does not and cannot have both position and momentum simultaneously." But originally, for the founders of QM, that was just an extension of the empirical criteria of meaning to the theoretical limit of the measurements of that day (and, as I understand it, was directly related to the technological limits of spectroscopes of that time).

That is simply incorrect. You will hear it a lot from people who have read too many superficial treatments of the history and haven't made the effort to (in Bhobba's apt phrase) "nut it out for themselves", but that doesn't make it correct.

In fact, the impossibility of making that simultaneous determination of position and momentum (or any other pair of non-commuting observables) appeared in the first mathematical formulations of QM. Heisenberg initially described this as "measuring one disturbs the other" but even then his argument was based on general principles and had nothing to do with technological limits.

I just mean careful, accurate, wise thinking about the principles that underlie science, and on questions such as the ones that we're discussing, and those that have become deeply intertwined in physics and particularly QM.

A necessary prerequisite for such thinking is to understand the way in which the uncertainty principle follows from the postulates of QM. Once you've done that, you can consider whether the problem lies in the postulates or in an error in the mathematical derivation from those postulates.

 

[Nugatory]

Bhobba has given OP a decent reading list. I'm closing this thread now.