I What Is the Role of Ontology in the Interpretations of Quantum Mechanics?

[MathematicalPhysicist]

"Measurement" is just an empty phrase of instrumentalists. Literally "it's the process in which something is measured".

Now seriously. Even most physicists who despise philosophy believe that the Moon is there when nobody observes it. (Mermin, for instance, doesn't believe it, but most physicists do.) And the statement that "the Moon is there when nobody observes it" is an ontological statement. So even physicists who proudly claim that ontology is a meaningless word - do have ontological beliefs. True, the word "ontology" is not precisely defined (except in the work by Harriagan and Spekkens), but it doesn't mean that it doesn't have any meaning at all.

In fact, in any scientific or mathematical theory it is impossible to give a precise definition of all words, simply because any definition requires a use of some words which also need to be defined, but one cannot have an infinite regress and circular definitions are not allowed. So some words must remain undefined, which are called primitive. For instance, in the standard minimal interpretation of QM which denies the existence of the measurement problem, the notion of measurement itself is primitive. This means that adherents of this interpretation think that the notion of measurement is clear intuitively, so that it doesn't need to be precisely defined. Likewise, adherents of some other interpretations (e.g. Bohmian or many-world) often think that the notion of ontology is clear intuitively, so that it doesn't need to be precisely defined. Similarly, a QBist may think that the notion of information is clear intuitively, etc. Different interpretations can therefore be reduced to what notion one thinks to be clear intuitively so that it can be considered primitive.

You are bound to have primitive objects.
In set theory the "set" is that primitive in Category theory "Category" is that object.

But you still have circularity, since those objects define themselves, you can't escape circularity and infinite regress in science and philosophy and also in life.

 

[A. Neumaier]

When Platonists answer "yes" to this question, what difference does it make? What expectations should I have about possible future events, that I would not have if the answer were "no"? Or, to put it another way, if you claim that numbers exist, I should be able to test that claim somehow. How would I test it?

I didn't claim that metaphysical statements can be empirically tested. Their purpose is to organize understanding, not to enhance predictability.

 

[A. Neumaier]

If I want to know where an electron was emitted, I design a source that tells me. If the source tells me an electron was emitted,

How does a source tell you this? Just because a manufacturer claims it does?

Now compare this with "numbers exist". How can I build a "number source" or "number detector" analogous to the electron source and electron detector above?

Any random number generator provides a source of numbers.

 

[hilbert2]

In Euclid's Elements there's an attempt to define things like points and lines, which in modern thinking make no sense to attempt to define. "A point is something that has no parts".

Maybe in some language there's a different word 'exist' for something that exists physically and something that exists as a concept. But I think the existence of numbers is a bit more concrete than that of many other things people imagine. There can be a surprisingly good agreement about properties of numbers, at least if you discuss them with someone who can understand them to the same level.

 

[lavinia]

One meaning of ontology in Physics is the study of the nature space. I believe this has a long history but perhaps the modern ideas began with Newton's argument for Absolute Space, a preferred frame of reference in which the laws of Physics hold. While I do not know the physics it would seems that this would include all inertial frames of reference - in the sense of Galileo. I believe that Leibniz argued against Absolute space and used philosophical arguments to show that Absolute Space should not exist.

Newton also believed in Absolute Time.

Here is an article on Leibniz's philosophy. Take a look at section 5 Leibniz on Space and Time.

https://plato.stanford.edu/entries/leibniz-physics/#AgaAbsSpaTim

The article says that Newton called Absolute Space the sensorium of God. Leibniz countered that since space has parts that would mean that God has parts and this is not possible.

One question I would have for physicists is whether the idea of space as a manifold is a modified idea of Absolute Space. It seems that gravity is not intrinsic to the manifold itself but is a metrical relation that depends of the distribution of matter and energy and which is imposed on the underlying manifold. So while there is no preferred frame of reference there is nevertheless an underlying space.

 

[lodbrok]

When Platonists answer "yes" to this question, what difference does it make? What expectations should I have about possible future events, that I would not have if the answer were "no"? Or, to put it another way, if you claim that numbers exist, I should be able to test that claim somehow. How would I test it?

Numbers exist. The proof of the claim is that we are talking about it. More precisely, the concept of numbers exists. Perhaps you are asking if numbers exist as entities independent of concepts, in which case that question won't make sense without a redefinition of what a number is, because in modern science, numbers are defined within an epistemological framework. You will have to come up with a framework in which numbers were ontological, and then you will be able to validly pose the question within that framework, and seek for ways of verifying the ontology of numbers, within the new framework.

Most of the time we spin our wheels asking questions that are meaningless within the framework we are using, in the sense that the premises of the questions contradict the core assumptions of the frameworks themselves.

BTW in case you are wondering, the answer is "yes", Bigfoot exists, ..., as a concept.

It isn't just a question about the use of words, it is a question about precision in what we mean.

 

[PeterDonis]

Numbers exist. The proof of the claim is that we are talking about it.

I can talk about unicorns. Does that mean unicorns exist?

More precisely, the concept of numbers exists.

That I agree with, but that is not the same as saying "numbers exist". The concept of unicorns exists, but that does not mean unicorns exist.

Perhaps you are asking if numbers exist as entities independent of concepts

I'm asking if numbers exist in the same sense that unicorns don't.

 

[lodbrok]

I'm asking if numbers exist in the same sense that unicorns don't.

Then the question is meaningless because by definition, numbers can't exist in the same sense as unicorns don't.

 

[PeterDonis]

How does a source tell you this? Just because a manufacturer claims it does?

I can test an "electron source" to see if whatever thingies it emits act like electrons.

Any random number generator provides a source of numbers.

Ok, this at least is a concrete answer. Yes, if you call particular bytes in a computer's memory or registers "numbers", then you have a number source.

 

[PeterDonis]

I didn't claim that metaphysical statements can be empirically tested. Their purpose is to organize understanding, not to add predictability.

Ok, this makes your position clearer. We can decide to call certain things we observe "numbers", just as we can decide to call certain things we observe "electrons".

 

[lavinia]

I think that the question of existence is deeply philosophical and is one of the main questions of philosophy historically. If I remember correctly Kant has a famous philosophical argument to show that existence is not a quality so it cannot be taken as a predicate.

https://philosophy.stackexchange.co...oes-kant-mean-by-existence-is-not-a-predicate

Given that, one needs to say what it means to exist.

To me mathematical objects transcend observation since for instance something like a sphere can never be observed. On the other hand mathematical objects certainly have properties such as curvature or dimensionality or algebraic structure. One studies them much as one studies an object detected by the senses. One examines them in order to determine their properties. One seeks general theories to describe classes of mathematical objects that share common properties. One seeks theories that unify and give insight into objects not yet examined. Much as among Physicists, many Mathematicians believe in a deep unity to all of Mathematics. It seems to me that the difference is that mathematics relies on proof for verification whereas Science relies on consistency with outcomes of experiment.

From a little reading, it seems that philosophy plays a key role in stimulating ideas in Science even though a priori philosophical claims are not scientific theories. Philosophy similarly has been key to mathematics.

 

[ftr]

Suppose numbers did not exist, would reality exist? Could there be a reality that is not describable by quantities?

 

[lavinia]

Suppose numbers did not exist, would reality exist? Could there be a reality that is not describable by quantities?

I think it is hard to say what it means that numbers exist because they can not be observed.

 

[ftr]

I think it is hard to say what it means that numbers exist because they can not be observed.

They either exist or they don't. Is there any other possibility?

 

[PeterDonis]

Suppose numbers did not exist, would reality exist?

What does "suppose numbers did not exist" mean?

@A. Neumaier explained what he means by "numbers exist". What do you mean by it?

They either exist or they don't. Is there any other possibility?

Yes: that "numbers exist" is not even a well-defined concept; it's just some words you strung together that don't mean anything.

 

[PeterDonis]

by definition, numbers can't exist in the same sense as unicorns don't.

What definition? By the definition @A. Neumaier gave, numbers (in his sense) do exist in the sense that unicorns don't.

 

[PeterDonis]

I think it is hard to say what it means that numbers exist because they can not be observed.

The "numbers" that @A. Neumaier defined can be observed.

 

[EPR]

The "numbers" that @A. Neumaier defined can be observed.

No. They 'exist' only within the framework of mathematics and have no separate, objective reality. A conscious tribesman who was never introduced to maths will never agree numbers exist. They only exist in a Platonist realm, so not in the same sense as electrons and matter.

Wikipedia:
Platonism at least affirms the existence of abstract objects, which are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness, and is the opposite of nominalism.[1] This can apply to properties, types, propositions, meanings, numbers, sets, truth values, and so on (see abstract object theory).

 

[vis_insita]

Yes: that "numbers exist" is not even a well-defined concept; it's just some words you strung together that don't mean anything.

Focussing on the phrase "numbers exist" is misleading. I believe the real question is whether statements like "There is an x such that for all y x is not the successor of y" mean something. If they do, then they must be either true or false. And if they are true, then there must be some thing referred to as "x" in this statement. The truth of such a statement is what is meant by "existence". Now there may be (in fact there are) a lot of different collections of things for which this statement is true. But If we decide to call one of those collections "the Natural Numbers", then natural numbers exist in a very meaningful sense. (A sense in which unicorns don't exist.)

 

[PeterDonis]

No

Go read his post again. You evidently have not.

 

[PeterDonis]

I believe the real question is whether statements like "There is an x such that for all y x is not the successor of y" mean something. If they do, then they must be either true or false.

No, such statements do not have a unique truth value, because you can get different truth values by picking different universal sets of which x and y are members. For example, if we say x and y are natural numbers, the statement is true; but if we say they are integers (i.e., they can be negative), the statement is false.

A better way of looking at statements like these is that they are assumptions that you can make in order to explore their implications. For example, we can assume that there is some set of objects for which your statement is true if x and y are members of that set; and we can then explore the properties of this set of objects. But that in no way means the statement is true for all sets of objects.

If we decide to call one of those collections "the Natural Numbers", then natural numbers exist in a very meaningful sense. (A sense in which unicorns don't exist.)

No, unicorns do exist in this sense, because I can define the concept of a unicorn as an object for which a certain set of statements is true, and by your definition, that is sufficient for such a concept to exist.

 

[vis_insita]

No, such statements do not have a unique truth value, because you can get different truth values by picking different universal sets of which x and y are members. For example, if we say x and y are natural numbers, the statement is true; but if we say they are integers (i.e., they can be negative), the statement is false.

It is irrelevant if the statement is false about some sets. All that matters is that it is true of some sets. At least one of the things of whatever is contained in any of the latter sets must exist, because that is what a true statement (by assumption) claims about it. And one of those sets contains all the natural numbers and nothing else.

As another example take the statement "There is an x, who wrote post #51 in this thread." It is also false about some sets of things. But the fact that it is true of, say, the set of all people living on planet Earth in 2020 is completely sufficient to establish the existence of PeterDonis, who is the unique individual who wrote that post in this thread.

A better way of looking at statements like these is that they are assumptions that you can make in order to explore their implications.

This is not a better way at all. It is, of course, completely legitimate to look at implications of statements, but it has nothing to do with what I am saying. Only the truth of certain statements (with regard to certain sets) is relevant for my point. And the truth of a statement has nothing to do with what statements it implies or by what statements it is implied, aside from the fact that true statements imply true statements.

For example, we can assume that there is some set of objects for which your statement is true if x and y are members of that set; and we can then explore the properties of this set of objects. But that in no way means the statement is true for all sets of objects.

Of course not. The only statements that are true for all sets of objects are tautologies. But I think that's besides the point.

No, unicorns do exist in this sense, because I can define the concept of a unicorn as an object for which a certain set of statements is true, and by your definition, that is sufficient for such a concept to exist.

No, it's not. My definition didn't allow you to arbitrarily define "unicorn". You are also not allowed to alter the definition of "first natural number" in any way that substantially differs from the one I gave. You have a point, though, that the term "unicorn" has some ambiguity. (So has the term "first natural number", but not in a way that affects its existence.) I was assuming that we would agree on a description of unicorns that implies that they would be some unusual assemblage of otherwise ordinary matter, or that they must have a position in space and time. Then they don't exist.

 

[Stephen Tashi]

and not "what" is actually there(ontology proper) as the ultimate ontology which we seek.

is foundation about ontology proper or not?

A mathematical approach would be to side-step all questions about the common language meaning of "exist" and "existence" and treat "things that exist" as an abstract set. Then we would state axioms that say if such-and-such is set of things that exist then so-and-so can be constructed from that set and also exists - or "exists" in some technically defined sense particular to the method of construction.

Expositions of physics don't obey this format!

The semi-philosophical issue of ontology might be clearer if we look at what "ontology" should mean in other branches of study - for example, economics and psychology. For example, most people agree that individual people exist. Does it follow that things like "love", "hate" and "paranonia" exist?

 

[PeterDonis]

@ftr Your argument amounts to choosing definitions of "exist" so that you are right and anyone who disagrees with you is wrong. Nothing you have said tells me why I should care about your definitions.

 

[ftr]

@ftr Your argument amounts to choosing definitions of "exist" so that you are right and anyone who disagrees with you is wrong. Nothing you have said tells me why I should care about your definitions.

Let me clarify. The argument for the nature of mathematics is usually put by philosophers/mathematicians/physicists as mathematics is either invented or discovered. Amounting to not exist, only because we humans have come up with it since it has no physical characteristics as we experience in reality, or it exist with the same status as physical reality since we can discover about its properties, although it is of different kind of existence.

So the question is well posed as far as many people who deal with the question. And both camps have their arguments and I am clearly in the later.

However, my original question

Suppose numbers did not exist, would reality exist? Could there be a reality that is not describable by quantities?

Was sort of proof by contradiction and to contemplate the question of a reality without any mathematical connection, it sounded like mind bending.

 

[ftr]

Expositions of physics don't obey this format!

Please see post #55 and the accompanying videos especially Penrose.

 

[PeterDonis]

So the question is well posed as far as many people who deal with the question. And both camps have their arguments and I am clearly in the later.

Then you should be able to show me a version of "numbers exist" that is well posed, as @A. Neumaier did. So far I haven't seen one from you.

 

[bhobba]

Let me clarify. The argument for the nature of mathematics is usually put by philosophers /mathematicians /physicists as mathematics is either invented or discovered.

Not quite - there are many views eg the view of Poincare and Wittgenstein that it is just a convention:
https://en.wikipedia.org/wiki/Philosophy_of_mathematics

Some may say that's invented, but that's the issue with philosophy - is following a convention inventing something? Its part of the reason we do not discuss philosophy here - but make a slight exception for some areas of quantum interpretations.

Thanks
Bill

 

[A. Neumaier]

I think it is hard to say what it means that numbers exist because they can not be observed.

Mathematicians have the notion of an existential quantifier to give a precise meaning to the notion of existence.

I believe the real question is whether statements like "There is an x such that for all y x is not the successor of y" mean something.

Mathematicians (who among all scientists have the most precise language) know how to give a precise meaning to all this. In the context of natural numbers, this uniquely specifies the smallest natural number (0 or 1, depending on whose conventions you follow).

Maybe in some language there's a different word 'exist' for something that exists physically and something that exists as a concept. But I think the existence of numbers is a bit more concrete than that of many other things people imagine.

Numbers are concepts like electrons, but the former's properties are much more familar to everyone than the latter's.

The pointer readings don't tell me how or where the electron was emitted, they tell me where the electron was detected. If I want to know where an electron was emitted, I design a source that tells me. If the source tells me an electron was emitted, yes, I will believe an electron was emitted even if I can't directly observe it--in short, I will believe that electrons exist.

We can decide to call certain things we observe "numbers", just as we can decide to call certain things we observe "electrons".

I can test an "electron source" to see if whatever thingies it emits act like electrons.

But this requires that you can measure something characteristic about electrons after they have been emitted. Which thingies do you call electrons?

[Numbers] 'exist' only within the framework of mathematics and have no separate, objective reality.

Electrons 'exist' only within the framework of physics and have no separate, objective reality.

 

[A. Neumaier]

One question I would have for physicists is whether the idea of space as a manifold is a modified idea of Absolute Space.

In classical relativity we have absolute space-time. In quantum gravity (for which we currently do not have a convincing theory) it is debatable whether or not spacetime is absolute; different approaches give different answers.