It is no more ‘located in the brain’ than actors are located inside televisions. Rather a rational mind is able to recognize such concepts which however are not dependent on being recognized in order to be real.

It is located in the brain just like simulated crocodile is located inside the computer and just like representations of actors are located inside a TV set. It's a virtual or mental kind of existence, a form of abstraction. i.e. representation.

What I'm aware of is this: every abstract idea is basically mined from the concrete. Your numerical example is perfect for demonstrating that: an idea pulled out of sets/collections of concrete objects. If so then zero must be an abstraction of sets that contain no members. Infinity, being more of a concept than an actual number, is to me, simply an extension of finite concrete sets; understandable in terms of the never-ending process of adding elements, say adding 1 to the preceding element, to a preexisting set.

You are referring to the concept of infinity that can not actually exist, by definition. The concept does not represent a thing to exist or not exist, nor it represents a collection or set of things. It represents an unfinished process of counting, and as such whatever it is accounting for can not exist because it is in perpetual state of becoming, just like you can not say a washing machine exists while its parts are still on assembly line and has not been put together yet. — Zelebg

A washing machine is assembled in ordered steps. If these are known then each step is a stage in the construction of the washing machine. Similarly the simple process of adding 1 to a preceding number as in 0,1, 2, 3,... is a buildup to infinity.

A washing machine is assembled in ordered steps. If these are known then each step is a stage in the construction of the washing machine. Similarly the simple process of adding 1 to a preceding number as in 0,1, 2, 3,... is a buildup to infinity.

Process describing a thing that will never be completed, that will never exist.

I say that such an assertion implicitly pre-supposes a division between the 'real world' which is presumed to exist independently of any conceptual framework, and the purported 'internal world' of ideas, concepts and abstractions, but that this division is really a false dichotomy. — Wayfarer

Yes, I assume that the images in our mind are different from the objects beyond our minds. Only you have access to your mind's contents; the content is subjective, and it comes to exist solely through mental activity, and ceases to exist when your mind ceases to exist (or sooner; you don't remember all the details of all your past experiences). This is different from objects of the external world, which exist independent of minds - they have objective existence.

And the problem with your theory of numbers, is that it fails to account for the 'unreasonable efficacy of mathematics in the natural sciences' — Wayfarer

The objects of the world are states of affairs whose constituents have relations to one another, and these relations can be described mathematically. I'm not saying the rekations don't exist, I'm just saying they don't exist independently of the states of affairs in which they are instantiated.

Does it really make more sense to suggest the relations exist independently, and the states of affairs that exhibit them have some sort of ontic relation to those platonic objects? The efficacy of math is not dependent on it.

Physics will never be able to prove the past is infinite, all it can possibly do is to show that no past boundary of time has been found. — Relativist

I was talking about spatial extension. Simply put, if space is infinite, which seems plausible from what we know, and if the rest of it looks much like what we can see around us, which is very plausible, then there's your actual infinity (if by that you mean an infinite number of material objects).

My argument is in the spirit of David Conway’s, in that I utilize the concept of completeness. However, Smith’s refutation doesn’t apply to my argument. — Relativist

How can an infinity of days become completed? — Relativist

You do not so much utilize the concept of completeness as just plug it in and expect it to do the work for you. Try to unpack the reasoning and you will see that it either does not apply or it begs the question against the existence of actual infinities.

Smith does address the sort of argument that you are hinting at in his section VI:

the collection of events cannot add up to an infinite collection in a finite amount of time, but they do so add up in an infinite amount of time. And since it is coherent to suppose that in relation to any present an infinite amount of time has elapsed, it is also coherent to suppose that in relation to any present an infinite collection of past events has already been formed by successive addition. — Smith, Infinity and the Past

(And he goes on to address Conway's and Craig's arguments in that vein.)

I’m not making the bold claim that an infinite past is logically impossible, I simply claim that there’s no conceptual basis for considering it POSSIBLE, and therefore it’s more rational to reject it. — Relativist

I fail to see the distinction that you are trying to draw here.

What I'm aware of is this: every abstract idea is basically mined from the concrete.Your numerical example is perfect for demonstrating that: an idea pulled out of sets/collections of concrete objects. If so then zero must be an abstraction of sets that contain no members. Infinity, being more of a concept than an actual number, is to me, simply an extension of finite concrete sets, understandable in terms of the never-ending process of adding elements, say adding 1 to the preceding element, to a preexisting set. — TheMadFool

Your suggesting that: since concrete objects entail abstract objects, that all abstract objects entail concrete objects. That does not follow; it commits the fallacy of affirming the consequent.

First order abstractions are formed by considering multiple objects of the world that have some common features, and mentally discarding the features that distinguish them. Second order abstractions are mentally constructed by extrapolation of first order absteactions - they don't necessarily have instantiations in the real world. Consider extrapolating from squares to cubes, to tesseracts, and beyond to higher dimensional analogues of cubes. They can only exist in the world if the world actually has that number of spatial dimensions.

I was talking about spatial extension. Simply put, if space is infinite, which seems plausible from what we know, and if the rest of it looks much like what we can see around us, which is very plausible, then there's your actual infinity (if by that you mean an infinite number of material objects). — SophistiCat

The first article showed that, according to accepted physics, spatial extension can only be infinite if there is an infinite past - that's why I focused on past time.

Smith does address the sort of argument that you are hinting at in his section VI:

the collection of events cannot add up to an infinite collection in a finite amount of time, but they do so add up in an infinite amount of time. And since it is coherent to suppose that in relation to any present an infinite amount of time has elapsed, it is also coherent to suppose that in relation to any present an infinite collection of past events has already been formed by successive addition.
— Smith, Infinity and the Past — SophistiCat

This does not apply to my claim. Sure, it's coherent - his statement entails no logical contradiction, but it circularly assumes the infinity exists, and it is that assumption that I challenge. The concept of infinity is a second order abstraction - an extrapolation of first order abstractions. (see the last paragraph of my last response to TheMadFool). A 90-dimensional analogue of a cube is an extrapolation of a cube. It could be described coherently, and consistent mathematical inferences could be made- nevertheless, this does not justify believing there exist objects of the world that correspond to it (there would actually have to exist 90 spatial dimensions). The general lesson is that we should be suspicious of second order abstractions - the mere fact that they have logicallly coherent properties does not establish their having real-world instantiations. Something more than logical coherence is needed to justify believing it.

I hope you see that my argument is purely epistemological, and specifically concerns whether or not belief in an actual infinity is justifiable.

The first article showed that, according to accepted physics, spatial extension can only be infinite if there is an infinite past - that's why I focused on past time. — Relativist

If you mean this article, then it doesn't say that, and it would be pretty incredible if it did, because the best established cosmological model to date implies an infinitely extended universe without the assumption of an infinite past. Note that the article spends considerable time talking about inflation, which is less well established than standard Big Bang cosmology, but inflation doesn't imply what you claim either.

This does not apply to my claim. Sure, it's coherent - his statement entails no logical contradiction, but it circularly assumes the infinity exists, and it is that assumption that I challenge. — Relativist

There is no circularity, because the refutation does not seek to establish the positive claim and makes no assumptions of its own. It only shows that the anti-infinitist argument doesn't work - the specific argument that you were making originally involving "completeness." (Morriston makes a similar case in Must the Past have a Beginning?)

As for your first order/second order abstractions, that's something completely different. I won't go further than just to say that I am not buying your epistemological construction.

If you mean this article, then it doesn't say that, and it would be pretty incredible if it did, because the best established cosmological model to date implies an infinitely extended universe without the assumption of an infinite past. Note that the article spends considerable time talking about inflation, which is less well established than standard Big Bang cosmology, but inflation doesn't imply what you claim either. — SophistiCat

The article states: "In fact, unless inflation went on for a truly infinite amount of time, or the Universe was born infinitely large, the Universe ought to be finite in extent." I admit I was only addressing the first possibility, but the second possibility remains just a brute fact assumption.

Physics analysis can't apply an arbitrary limit, so they must start with an absence of boundaries. That does not establish the existence of an actual infinity, it just indicates that the physics entails no boundary.

As for your first order/second order abstractions, that's something completely different. I won't go further than just to say that I am not buying your epistemological construction. — SophistiCat

At least you see it's different. I consider rational belief to require rational justification, and logical consistency seems inadequate as a rational justification. 90-dimensional cube analogues are logically consistent, but there's no rational basis for believing they exist in objective reality.

...if by that you mean an infinite number of material objects

Infinity is not a number. No logically valid statement can account for uncountable.

the collection of events cannot add up to an infinite collection in a finite amount of time, but they do so add up in an infinite amount of time. And since it is coherent to suppose that in relation to any present an infinite amount of time has elapsed, it is also coherent to suppose that in relation to any present an infinite collection of past events has already been formed by successive addition.
— Smith, Infinity and the Past

Starts with fallacy, continues with assumption, ends with paradox.

The article states: "In fact, unless inflation went on for a truly infinite amount of time, or the Universe was born infinitely large, the Universe ought to be finite in extent." I admit I was only addressing the first possibility, but the second possibility remains just an assumption. — Relativist

As is the alternative. It is widely understood that the universe (or at least the inflationary "pocket" that we occupy, if we go with the still somewhat speculative eternal inflation hypothesis) was either always finite or always infinite. Neither option is granted the default status by cosmologists (your epistemological arguments notwithstanding). The question is decided by the weight of evidence, or lacking that, by considerations of simplicity; these two epistemological criteria constitute the foundation of scientific epistemology. As far as the former, there have been some publications that claimed that a closed (finite) spacetime topology fits some of the observational evidence better than the flat topology of the standard FLRW model, but their conclusions have not yet gained widespread acceptance. As for simplicity, here the flat semi-Euclidean spacetime wins over the more structurally complex closed topologies.

Abstractions like triangles are well defined, that's why we can each consider objects with triangular properties, and engage in the way of abstraction. That does not entail the independent existence of the abstraction, "triangle". — Relativist

But what is the status of the definition? I would have thought that triangles would be discovered in all possible worlds, in other words, their reality is not dependent on our definition of it, but our definition of it must conform to the concept (of a flat plane bounded by three intersecting straight lines). The issue is that your notion of 'existence' is too narrow; it comprises only physical objects, actual triangles, which you assume are alone 'real objects'. But geometrical concepts, such as triangles, are first and foremost concepts, of which physical forms are but representations. So rather than triangles being physical objects represented by ideas in the mind, the idea 'triangle' is a concept represented by physical examples.

I assume that the images in our mind are different from the objects beyond our minds. — Relativist

We know of anything beyond the mind through the processes of assimilation and synthesis, which takes place in the mind. We have a picture of the world, a master-construct if you like, of the domain of objects with you, the subject, within it. But this doesn't acknowledge the sense in which this construction of self-and-world is also the work of the mind (referred to in, for example, Schopenhauer's philosophy as 'vorstellung'. ) So again, in everything you write, you're assuming a reality independent of any observing mind, but that assumption might be philosophically problematical, i.e. it might not be as self-evident as you're assuming it to be.

Does it really make more sense to suggest the relations exist independently, and the states of affairs that exhibit them have some sort of ontic relation to those platonic objects? — Relativist

I believe so. Don't overlook the fact that the current model of the so-called fundamental constituents of matter, the so-called 'particle zoo', is a mathematical model. Certainly it has to be validated against empirical data, but it is arguably a purely intelligible construct (which is close to what some physicists say about it.) Have a look at this paragraph from the SEP entry on mathematical platonism.

Your suggesting that: since concrete objects entail abstract objects, that all abstract objects entail concrete objects. That does not follow; it commits the fallacy of affirming the consequent.

First order abstractions are formed by considering multiple objects of the world that have some common features, and mentally discarding the features that distinguish them. Second order abstractions are mentally constructed by extrapolation of first order absteactions - they don't necessarily have instantiations in the real world. Consider extrapolating from squares to cubes, to tesseracts, and beyond to higher dimensional analogues of cubes. They can only exist in the world if the world actually has that number of spatial dimensions. — Relativist

I see. If I understood you correctly then there are such things as second-order abstract objects e.g. tesseract. However zero and infinity are first-order abstractions in my opinion; so they should have concrete instantiations.

Process describing a thing that will never be completed, that will never exist. — Zelebg

Why do you say that? A process that has begun represents the process right?

Neither the process nor that it has begun was focus of my point, but infinite set the process is assembling and the fact that it can never be complete, by definition. Infinite set will never be complete, therefore infinite set will never exist. There is no such thing as infinite past, it's simply a self-contradiction.

Neither the process nor that it has begun was ficus of my point, but infinite set the process is assembling and the fact that it can never be complete, by definition. Infinite set will never be complete, therefore infinite set will never exist. There is no such thing as infinite past, it's simply a self-contradiction. — Zelebg

Of course it can't be completed but it can be started. Look: 0, 1, 2, 3, 4, 5,...

Are you a robot?

Infinite set will never be complete, therefore infinite set will never exist.

I agree it can't be completed and my intuition on the matter may be off the mark but consider this: I sometimes see only part of a person, say when that person is behind a tree or low wall but that doesn't mean the person doesn't exist does it?

Hence I continue to disagree with you (and McTaggart) that time is an unreal illusion. — aletheist

Okay. I still think when you do the simple phone call across the globe and/or 4th dimensional time travel, it creates the paradox from present tense. For instance, when traveling from west to east, another paradox presents itself by virtue of one being unable to live the lost hours (or in the opposite you get to relive them).

Nonetheless, the video synopsis could be summed up in the simple statement:

1. Eternity is Time. Time, eternity.

Is that true or false?

(Some things we covered)
a. it's an abstract
b. it's a reality
c. it's an illusion
d. it's all of the above
e. it's something else

If time is illusion how do you explain velocity and motion in general?

Hi Zelebg!

Generally, you could explain it through a simple mathematical formula (time= distance/speed). And as such, we are back to abstracts. Mathematical abstracts.

Are they real? (Where do they come from, a priori.) How is this phenomenon even possible?

These are intriguing questions for sure... !

I see. If I understood you correctly then there are such things as second-order abstract objects e.g. tesseract. However zero and infinity are first-order abstractions in my opinion; so they should have concrete instantiations. — TheMadFool

Zero is second order: it is conceived as a negative fact, like removing the apples from a basket, one by one, ultimately leaving 0 apples. Negative facts are indirect - they don't tell us what IS, they tell us a subset of what isn't. One refers to 0 apples in this example only because of the psychological context - we're considering apples. Although the basket has no apples, it may contain oranges and orangutans. It is the indirect nature of negative facts that makes them second order.

I agree it can't be completed and my intuition on the matter may be off the mark but consider this: I sometimes see only part of a person, say when that person is behind a tree or low wall but that doesn't mean the person doesn't exist does it? — TheMadFool

Consider how the notion of infinity is manifested in your counting example. Counting is a process. There is no infinity at any identifiable step of the process. Rather, infinity manifests as the process itself, one that never ends - and process is not an existent. It's still a reasonable way to conceive of a potential infinity, but this conception doesn't work for an actual infinity - including a past infinity. A conception for an actual infinity cannot be some ongoing process. It would entail a COMPLETED process.

Generally, you could explain it through a simple mathematical formula (time= distance/speed).

speed = distance / time

So now I explained how time is in fact not an illusion. What the hell am I doing here when I have more meaningful conversations with my dog.

Hahaha, in a strange way...kind of reminds me of the pyramids at Giza... we only figured that out, abstractly :brow:

Does it really make more sense to suggest the relations exist independently, and the states of affairs that exhibit them have some sort of ontic relation to those platonic objects?
— Relativist

I believe so. Don't overlook the fact that the current model of the so-called fundamental constituents of matter, the so-called 'particle zoo', is a mathematical model. — Wayfarer

Mathematical models don't entail the independent existence of platonic entities.

Consider the simple case of the relations between electrons and protons. How do we account for these with platonic entities? We'd have an electron with some sort of ontic relation to the abstract object "-1 electric charge", and the proton has an ontic relation to the abstract object "+1 electric charge", and then these abstractions have an "attraction relation" between them. It is simpler to simply assume the attraction relation exists between electrons and protons by virtue of their respective intrinsic properties (their electric charge). This account minimizes the number of ontic entities, and does not constrain our ability to reason and do math.

what is the status of the definition? I would have thought that triangles would be discovered in all possible worlds, in other words, their reality is not dependent on our definition of it, but our definition of it must conform to the concept

The definition of triangle is not impacted, just the ontology behind it. There are states of affairs that have triangularity among the constituents. We can mentally consider the property "triangular" while ignoring the other details of the object, but that doesn't require the abstract object "triangle" to be ontic. It's a semantic convenience, and has the utility of allowing us to do the math,but we can do the math witthout making an ontological commitment to them.

, you're assuming a reality independent of any observing mind, but that assumption might be philosophically problematical, i.e. it might not be as self-evident as you're assuming it to be. — Wayfarer

Yes, I'm assuming a reality independent of the observing mind; i..e. I deny solipsism. But this belief is not a deduction, it's just an articulation of a component of our intrinsic (not deduced) view of the world. While it's possibly false, that mere possibility is not adequate grounds to undercut the belief in an external world - a belief that is directly derived from considering our hard-wired relation to that world.

This is about ontology- what actually exists, and we should not be promiscuous with our assumptions about what exists. Triangularity still exists, and we can still reason with triangles (as well-defined mental objects) without adding them to the furniture of the world.

For instance when traveling from west to east, another paradox presents itself by virtue of one being unable to live the lost hours (or in the opposite you get to relive them). — 3017amen

No, there are no "lost" or "relived" hours. That is an illusion created by our arbitrary manner of marking and measuring time.

Nonetheless, the synopsis of that video could be summed up in the simple statement: Eternity is Time. Time, eternity. — 3017amen

Where are you getting that from the video? For one thing, the word "eternity" is never mentioned.

Is that true or false? — 3017amen

I cannot answer until you elaborate on what you mean by it. In any case, I have just started a new thread on "The Reality of Time" and suggest that we continue this conversation over there.

you could explain it through a simple mathematical formula (time= distance/speed). And as such, we are back to abstracts. Mathematical abstracts.

Are they real? (Where do they come from, a priori.) How is this phenomenon even possible? — 3017amen

Distance exists in the real world as an ontic relation between two objects (two states of affairs) separated in space. Time exists as an ontic relation between two events (two states of affairs) separated in time. Considered apart from their respective objects, distance and time are abstractions - mental objects, that can be abstractly mapped to numbers, which we can mentally manipulate mathematically - computing such things as ratios (distance/time = speed).

Consider the simple case of the relations between electrons and protons. How do we account for these with platonic entities? — Relativist

Again, your definition of what exists is too narrow. It's restricted to objects. But mathematical relationships and ratios also exist - it's simply that the don't exist as objects, but are implicit in our understanding of what objects are, and how they relate to each other. So saying that numbers and the like are real, is not saying that they exist as literal objects, but as they're intrinsic to rational thought then they're real in a way that literal objects are not.

I'm assuming a reality independent of the observing mind; i..e. I deny solipsism. — Relativist

I can see we're not going to clear this hurdle, but for what it's worth: in referring to the reality of mind, I'm not referring to your mind, or my mind, or any individual instance of consciousness. I'm referring to the vast expanse of shared meanings within which all discourse about objects is conducted. You think that what 'really exists' are only the objects of perception within this matrix of ideas, but I'm arguing that it's the matrix of ideas that is real. (This is near in meaning to the ancient Greek term 'nous'.)

But the underlying issue is a very deep one: modern science (and so much of modern thought) starts with the methodological assumption of a mind-independent world; which for the purposes of science, is a valid assumption. But it has no ultimate reality, because reality itself is not something that we're outside of, or apart from. This is something that has started to become clear in recent science in multiple ways (see for instance this blog post.)