Infinity and Zero: do they exist?

Relativist

Abstractions do not exist (not in the real world), they are mental devices that we create via The Way of Abstraction. They reflect types of objects that share some common properties. For example, the abstract object triangle does not actually exist (not in the real world) but triangular objects exist - they are instantiations of the abstraction.

In mathematics and logic, we use the existential quantifier(∃) in a way inconsistent with the above, often applying it to abstractions. But that doesn't confer real world existence to the abstraction.

Numbers are abstractions, some of which are instantiated in the real world. 4 is instantiated as a property of any group of four objects (properties exist, but only as constituents of the objects in which they are instantiated). π Is not instantiated; it is an irrational number - a ratio of circumference to diameter - a second level property, not directly instantiated - it is a matematical relation between abstract objects.. It's a useful abstraction but it's still just an abstraction.

Zero is another useful abstraction. 0 has mathematical relations to other numbers, but it is not instantiated in the real world. It refers to an absence of existence, not to an existent.

Now we get to infinity. The arguments I've seen for its existence (e.g. referring to Cantor) only show that it is a logically coherent, mathematically useful, abstraction (like π, or 0). This does not establish its existence in the real world - an instantiation.

While this doesn't prove the nonexistence of infinity, it does undercut proofs that infinity does exist. At best (for infinity-o-philes) , we should reserve judgment. But I submit it's reasonable to go further, and reject its existence just as we should reject the (actual, real-world) existence of π and 0, for lack of good reasons to believe otherwise.

3017amen

↪Relativist

Great question R !!

In a word, I want to say yes. They exist as metaphysical abstracts.

Common analogies (that underlie/describe the physical world), which you probably already know, include:

1. Laws of gravity in physics
2. Engineering/Design formulas for; compressive forces, tensile strength, torsional forces, etc.
3. Musical notation

Also, I agree, I think Omega is useful, yet unknowable.

My question would be how would those metaphysical abstract's relate to the phenomenal world of consciousness and Being, if at all(?). In other words, how useful are they to us, in that context of consciousness and Being. In a paradoxical way, numbers and math can describe the physical world's underlying properties, yet they are purely abstract. I wonder if there is a connection to something...

Relativist

Laws of gravity in physics
2. Engineering/Design formulas for; compressive forces, tensile strength, torsional forces, etc.
3. Musical notation — 3017amen

In my view:
Laws of physics are relations between types of things. Things can relate to one another in ways that can be described mathematically. That doesn't entail independent existence apart from the things that relate in that way.

Same thing essentially applies to engineering formulae- they still are due to natural law.

Musical notation isn't an abstraction, it's a semantics that maps to various aspects of sound.

You refer "metaphysical abstracts" - suggesting abstractions actually exist as (what?) Platonic entities? Why think they exist in this way rather than merely as a mental entity, formulated via the Way of Abstraction?

3017amen

That doesn't entail independent existence apart from the things that relate in that way. — Relativist

I'm thinking that it would entail an independent existence. If a given abstract does not exist for the sole purposes of the creation of a particular concrete thing, by definition, it would then be something independent of the thing itself. In other words, if it can be created and/or exists without such abstract knowledge, then what is its purpose for existing?

To that end, using the laws of gravity once more, they do not confer any biological survival value when one can simply dodge falling objects.

Musical notation isn't an abstraction, it's a semantics that maps to various aspects of sound. — Relativist

I'm not sure I'm with you on that one. Much like written mathematical formula, musical notation represents otherwise-produced symbols, including notation for durations of sound as well as the absence of sound such as rests. I don't see the difference, or even how semantics would play a role.

As far as Platonic abstracts, in my view, the link to timeless truth's is about the closest metaphor we can relate to (in order to be useful), when comparing universal's such as mathematical abstracts; I think they can be non-physical phenomena, yet dependent on consciousness for their apprehension. I'm not a Platonist, but maybe Wayfarer could elucidate there, since Platonism would include the idea that both non-physical and non-mental abstracts exist.

But back to abstracts music and math. When looking at a pre-engineered beam, I know what exists abstractly is a mathematical formula that underlies its physical reality. Conversely, when I read abstract musical notation, I hear music. Abstracts can work both ways.

GrandMinnow

Now we get to infinity. The arguments I've seen for its existence — Relativist

In set theory (since you mentioned Cantor, who provided the main pre-formal concepts) there is not an entity called 'infinity' (distinct from a other notions such as points on a real extended line or figures of speech such as "as x goes to infinity"). Rather, there is the adjective 'is infinite', and an axiom that entails (with other axioms) certain theorems including the existence of infinite sets.

GrandMinnow

In mathematics and logic, we use the existential quantifier(∃) in a way inconsistent with the above — Relativist

Existential quantification is not inconsistent with the claim that abstractions are not material objects.

GrandMinnow

reject its existence — Relativist

Of course, we can hold that there do not exist infinite sets. But then providing a formal axiomatization for the mathematics for the sciences gets a lot more complicated.

Relativist

If a given abstract does not exist for the sole purposes of the creation of a particular concrete thing, by definition, it would then be something independent of the thing itself. — 3017amen

That sounds like Platonism. My problem with ontologies that include platonic objects is that they seem unnecessary. Why posit an independent existence for triangles, when triangles can be accounted for as constituents of triangular objects? Further, how do triangles exist independently? How do they get connected to objects? Can the connection be severed? This makes it even more unnecessarily complex? Can they replaced with squares simply by replacing the connection?

. I don't see the difference, or even how semantics would play a role. — 3017amen

The notation is interpreted by a musician, analogously to a reader interpreting print words. Words refer to objects, concepts, actions etc, while musical notations refer to the various aspects of sounds you mention. The sounds can be reproduced on an instrument, or merely interpreted within the musician's mind.

Abstracts can work both ways. — 3017amen

Both ways are consistent with the way of abstraction. We mentally consider a set of attributes common to all triangles to form the abstraction in our minds, then reverse the process, adding back concrete elements.

Relativist

Of course, we can hold that there do not exist infinite sets. But then providing a formal axiomatization for the mathematics for the sciences gets a lot more complicated. — GrandMinnow

Sets are abstractions. Creating abstraction just means conceptualizing. My point is that abstractions don't actually exist except as mental entities. Mathematical abstractions are useful because they entail analyzable properties Does anyone suggest imaginary numbers exist? Nevertheless, they appear in physics equations.

3017amen

That sounds like Platonism. My problem with ontologies that include platonic objects is that they seem unnecessary. Why posit an independent existence for triangles, when triangles can be accounted for as constituents of triangular objects? Further, how do triangles exist independently? How do they get connected to objects? Can the connection be severed? This makes it even more unnecessarily complex? Can they replaced with squares simply by replacing the connection? — Relativist

I don't think it's Platonism because it assumes an independent existence outside of consciousness. The triangulation of a roof truss exists abstractly. The connection can be 'severed' and independent of the concrete thing itself, the roof truss.

The notation is interpreted by a musician, analogously to a reader interpreting print words. Words refer to objects, concepts, actions etc, while musical notations refer to the various aspects of sounds you mention. The sounds can be reproduced on an instrument, or merely interpreted within the musician's mind. — Relativist

Yes, what you stated is the phenomenon relative to metaphyscal language.

Both ways are consistent with the way of abstraction. We mentally consider a set of attributes common to all triangles to form the abstraction in our minds, then reverse the process, adding back concrete elements. — Relativist

Yes, I agree. But they [abstracts] are not needed to build concrete things, that exist.

Relativist

Existential quantification is not inconsistent with the claim that abstractions are not material objects. — GrandMinnow

Agreed. The question remains: do immaterial objects exist? If so, what does it mean to exist? Does Spider-Man exist? Do all fictions, past present, and future exist? What about possible fictions that never get authored?

Relativist

There is not an entity called 'infinity' (distinct from a different notion of points on a real extended line or figures of speech such as "as x goes to infinity"). Rather, there is the adjective 'is infinite', and an axiom that entails (with other axioms) that entails certain theorems including the existence of infinite sets. — GrandMinnow

Fair point, although infinites appear in some physics equations, and they are treated ad objects in transfinite math. Regardless, from this viewpoint, the question is: is there something that exists in the real world that maps to an infinite set?

3017amen

Do all fictions, past present, and future exist? — Relativist

Think of time itself, as being both abstract and concrete. (Does time exist? And how does it exist, abstractly?)

Relativist

↪3017amen

Time exists a a relation between states of affairs. I don't believe abstraction exist independently of states of affairs.

GrandMinnow

↪Relativist

The presence of such questions doesn't impugn existential quantification.

GrandMinnow

↪Relativist

You want mathematics not to claim that there exist infinite sets; you want to "reject existence", as I understood you, such as existence asserted with the existential quantifer. But you haven't answered my point that without infinitistic set theory, axiomatizing the mathematics for the sciences gets a lot more complicated.

Relativist

I don't think it's Platonism because it assumes an independent existence outside of consciousness. The triangulation of a roof truss exists abstractly. The connection can be 'severed' and independent of the concrete thing itself, the roof truss. — 3017amen

OK, but that's just referring to a concept - a mental object. It is spatially located in your brain, unless dualism is true. Triangular objects exist even if there are no minds to conceptualize triangles. When people speak of the existence of infinity they are not merely referring to the concept that exists in our minds.

Relativist

That doesn't answer my point that without infinitistic set theory, axiomatizing the mathematics for the sciences gets a lot more complicated. — GrandMinnow

Sorry. I agree with that. They are useful fictions.

Relativist

The presence of such questions doesn't impugn existential quantification. — GrandMinnow

No, but one shouldn't conflate existential quantification with a statement of ontology. IOW just because we can do some useful math with infinities doesn't entail anything ontic.

GrandMinnow

the question is: is there something that exists in the real world that maps to an infinite set? — Relativist

That raises the question, "What do you mean by 'the real world'"? And what do you mean by "something exists in the real world"?

Anyway, whatever the answer about infinite sets, mathematical statements themeselves are things like "There exists the set of natural numbers" or "There exists an x such that, for all y, if y is a natural number then y is an element of x" or "With these axioms we prove the formula 'There exists the set of natural numbers'" and not "There exists the set of natural numbers in the real world (whatever "in the real world" might mean)."

GrandMinnow

↪Relativist

I don't opine on that particular philosophical position. But the outcome of it doesn't preclude existential quantifcation in mathematics or working with infinitistic set theory.

GrandMinnow

They are useful fictions. — Relativist

I don't opine whether they are fictions or not. But, for me, at least such views as fictionalism and instrumentalism that allow sets as consistent fictions is enough for working in mathematics.

Relativist

That raises the question, "What do you mean by 'the real world'"? And what do you mean by "something exists in the real world"? — GrandMinnow

Yes, and thus we get into metaphysics. A topic for another day.

3017amen

Triangular objects exist even if there are no minds to conceptualize triangles. — Relativist

How is that possible?

3017amen

Time exists a a relation between states of affairs. I don't believe abstraction exist independently of states of affairs. — Relativist

What does state of affairs mean?

Relativist

↪3017amen

Everything that exists is a State of Affairs. It's constituents are; a particular, its attached properties, and its relations to other states of affairs. This acknowlwdges that properties exist only in their instatiations in a state of affairs.

Relativist

Triangular objects exist even if there are no minds to conceptualize triangles.
— Relativist

How is that possible? — 3017amen

A triangular object has 3 sides that are arranged in a certain general way. It's existence and structure is not dependent on a mind analyzing that structure.

3017amen

Everything that exists is a State of Affairs. It's constituents are; a particular, its attached properties, and its relations to other states of affairs. This acknowlwdges that properties exist only in their instatiations in a state of affairs. — Relativist

In a humanistic sense, are you saying that we all are an interconnected consciousness?

3017amen

A triangular object has 3 sides that are arranged in a certain general way. It's existence and structure is not dependent on a mind analyzing that structure. — Relativist

Isn't that Platonism?

Relativist

↪3017amen

No, because the structural properties exist only in the objects that have them. Platonism would entail their existence independent of those objects.

Relativist

In a humanistic sense, are you saying that we all are an interconnected consciousness? — 3017amen

Not really. The relations between consciousnesses seems indirect.

Infinity and Zero: do they exist?

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