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

[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.

 

[PeterDonis]

Which thingies do you call electrons?

The thingies that have the properties of electrons.

 

[A. Neumaier]

The thingies that have the properties of electrons.

But which observable properties do they have, given that one can observe nothing but pointer readings?

 

[vis_insita]

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

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).

I completely agree. The point I was trying to get at is this: Consistently applying the precise meaning that mathematicians give to the notion of existence, enables us to talk about "ontology" in a way that is completely free of vagueness. We just have to (in principle) state which existentially quantified statements we believe to be true. (Promising candidates I imagine to be statements taken from our best scientific theories and mathematics.) This makes it easy to have an ontology which contains numbers and electrons, but not unicorns. We just declare some appropriate set of arithmetical statements (like the Peano axioms) to be true and statements implying "There exists an x, such that x is a <precise description of unicorns>" to be false.

My question regarding the meaning of a particular arithmetical statement was directed generally at people who say that numbers don't exist. There are only three possibilities for any set of arithmetical statements

1) It is meaningless
2) it is false of everything (contradictory)
3) it is true of something.

The latter case 3) applied to, e.g the Peano axioms, I think, inevitably implies that numbers exist, because it implies "There exists an x, such that for all y x is no successor of y", which is true if and only if there exists something that is not the successor of any other thing. And one of those things for which this statement is true I can as well call "The smallest natural number." This is why I am interested to know whether people denying the existence of numbers believe all arithmetical statements to be meaningless. (I think they must.)

 

[PeterDonis]

which observable properties do they have, given that one can observe nothing but pointer readings?

The set of observable properties that we name "electron".

 

[A. Neumaier]

The set of observable properties that we name "electron".

''electron'' is not the name of a set of observable properties, but the name for a concept from theoretical physics (refining much older less precise concepts from experimental physics).

Observable are positions, currents, spectra, cross sections, decay rates, detector correlations - nothing that would characterize an electron.

 

[PeterDonis]

''electron'' is not the name of a set of observable properties, but the name for a concept from theoretical physics

And that concept from theoretical physics ultimately leads you back to observations: the particular observations that convinced theorists that "electron" was a concept that needed to be included in their theories, and the observations that continue to convince physicists to use those theories because they make accurate predictions.

Earlier you said that bytes stored in a computer's memory were "numbers"; that claim is open to all the same objections you are making against my statements about electrons. So now I'm confused about your position. I thought you were saying that "numbers exist" because we can observe them in the memories of computers, and similarly "electrons exist" because we can observe them in the measurements that confirm our theories. I have no objection whatever to that position. But now you seem to be arguing against it.

 

[lavinia]

And that concept from theoretical physics ultimately leads you back to observations: the particular observations that convinced theorists that "electron" was a concept that needed to be included in their theories, and the observations that continue to convince physicists to use those theories because they make accurate predictions.

So it seems that electrons exist as a concepts. But so do spheres and numbers.

There are concepts in Mathematics that lead back to further investigation and some concepts lead to new and unexpected insight. For instance, De Rham's theorem leads to the theory of differential extensions of cohomology theories.

 

[A. Neumaier]

And that concept from theoretical physics ultimately leads you back to observations: the particular observations that convinced theorists that "electron" was a concept that needed to be included in their theories, and the observations that continue to convince physicists to use those theories because they make accurate predictions.

Earlier you said that bytes stored in a computer's memory were "numbers"; that claim is open to all the same objections you are making against my statements about electrons. So now I'm confused about your position. I thought you were saying that "numbers exist" because we can observe them in the memories of computers, and similarly "electrons exist" because we can observe them in the measurements that confirm our theories. I have no objection whatever to that position. But now you seem to be arguing against it.

I don't see this analogy as that close. We never observe electrons but only their effects on detectors predicted by theory, while we explicitly manipulate numbers all the time. Thus numbers exist for much more elementary reasons than electrons. (The random number generator was not the answer to ''numbers exist'' but to your quest for a "number source" in post #30.)

How can I build a "number source" or "number detector" analogous to the electron source and electron detector above?

I was arguing against your comment in #30 to the last question in my statement

Though the term ''reality'' may have multiple meanings (and hence needs a philosophical analysis to disentangle their various uses), it is a term needed - even though people like @vanhees71 substitute it by phrasing things in terms of ''observational objective facts'' rather than ''reality''. But this compound term is not really less ambiguous. Are electrons factually emitted by a source though we only observe pointer readings?

My implied answer to this rhetorical question was: ''of course'', while you put it in doubt.

The ''observational objective facts'' imply the factual existence of electrons (as instances of the concept of excitations of the electromagnetic field), though they are not directly observable but only inferred through consistency of their manipulation in theory and experimental practice.

Similarly, numbers factually exist (as instances of the concept of a complex number or one of its special cases), because of the consistency of their manipulation in theory and computational practice.

This is the true analogy regarding existence of electrons and numbers.

 

[PeterDonis]

We never observe electrons but only their effects on detectors predicted by theory, while we explicitly manipulate numbers all the time.

Really? You explicitly manipulate the individual bytes in your computer's memory? You directly observe them?

Your evidence for the existence of "numbers" in the sense you defined them is just as indirect as my evidence for electrons.

 

[PeterDonis]

I was arguing against your comment in #30 to the last question in my statement

My implied answer to this rhetorical question was: ''of course'', while you put it in doubt.

Evidently I didn't make myself clear. In post #30 I wasn't questioning the existence of electrons, I was questioning the existence of numbers. Then you explained how to make a "number source"--just use your keyboard or mouse or a combination of them to cause your computer to store certain bytes in its memory, which you can then manipulate in ways that match the definition of numbers. I accept this as a "number source".

However, you then need to be consistent in your claims about numbers "existing". If there is no computer anywhere that stores bytes in its memory that can be manipulated in ways that match the definition of numbers, then numbers don't exist. Just as if there were no objects anywhere in the universe that could produce the observational objective facts that imply the factual existence of electrons, then electrons would not exist.