On the Existence of Abstract Objects

Let’s begin with perception. I experience the physical world though my five senses: sight, taste, touch, hearing, and smell. I do not possess a special “tree-sensing” sense. So how can I experience a tree? The answer is I do not directly experience the tree.

An abstract object is defined as something which is neither spatial nor temporal: an abstract object does not exist in space and time. (“Object” should not be taken too literally; think “abstract entity.”) A typical example of abstract objects is numbers. Numbers such as 2 or ? do not exist in space/time. Yes, two apples exist in a particular place at a particular time; but the number 2 itself does not.

We can directly see on only one thing: light. The mind does the rest. Almost everything we experience though our senses are universals, are abstract objects, are ideas in our mind.

My mind directly experiences the number two because the number two is a thought and my mind experiences thoughts directly. Similarly, my mind can directly experience the abstract object named “tree” because that, too, is a thought. As to what is causing my experiences, I suppose there’s a material object, a material tree, existing in a physical universe outside myself. If I’m not dreaming, hallucinating, or a brain in a vat, then my supposition may be correct. There may actually be a material tree, existing in a physical universe outside myself. But, then again, there might not.

If nearly all the words found in the dictionary stand for abstract objects, then it follows that our experience consists almost exclusively of abstract objects, as I’ll show.

So how can I experience a tree? The answer is I do not directly experience the tree.

We can directly see on only one thing: light. The mind does the rest.

A “brain in a vat” experiences rough brown patches and smooth green patches. Of that experience the brain can be absolutely certain. The brain naturally supposes something in the physical universe called a “tree” is the cause of what we experience. It is wrong.

My mind directly experiences the number two because the number two is a thought and my mind experiences thoughts directly. Similarly, my mind can directly experience the abstract object named “tree” because that, too, is a thought. As to what is causing my experiences, I suppose there’s a material object, a material tree, existing in a physical universe outside myself. If I’m not dreaming, hallucinating, or a brain in a vat, then my supposition may be correct. There may actually be a material tree, existing in a physical universe outside myself. But, then again, there might not.

Instead of questioning the existence of abstract object, perhaps we should apply our skepticism the existence of an exterior physical universe.

Let’s begin with perception. I experience the physical world though my five senses: sight, taste, touch, hearing, and smell. I do not possess a special “tree-sensing” sense. So how can I experience a tree? The answer is I do not directly experience the tree. Rather, my eyes see patches of brown and green; my fingers tell me the brown patches are rough and the green patches are smooth. My mind retrieves the idea “tree” to explain what my senses are telling me. Tree is a mental representation which describes what I experience. (We may suppose a newborn infant only sees patches of light. Over time, the infant deduces the ideas of object, object permanence, and eventually tree.)

I experience the physical world though my five senses: sight, taste, touch, hearing, and smell. I do not possess a special “tree-sensing” sense. So how can I experience a tree? The answer is I do not directly experience the tree. Rather, my eyes see patches of brown and green; my fingers tell me the brown patches are rough and the green patches are smooth. My mind retrieves the idea “tree” to explain what my senses are telling me. Tree is a mental representation which describes what I experience. (We may suppose a newborn infant only sees patches of light. Over time, the infant deduces the ideas of object, object permanence, and eventually tree.)

An abstract object is defined as something which is neither spatial nor temporal: an abstract object does not exist in space and time.

I have no further comment on abstract objects, but I do have something to say about all these purported "illusions" the mind creates. . . . Secondly, the generalization from such "illusions" is dubious -

This way however, we may at least indirectly experience an abstract tree - through experience of concrete objects and their concrete causal associations in the brain.

Our five physical senses limit us to experiencing sight, sound, taste, touch, and smell. So, we don’t directly experience concrete objects.

What could this mean “we don’t directly experience concrete objects” I see a tree, I go over to touch the leaves, smell the bark, hear the creaking of the branches, or taste the fruit it produces. How more direct can we get?

Austin's criticism of "universal" concepts starts at about p.84.

As a matter of fact, if any one were anxious to deny altogether that there are such things as universals, we should find that we cannot strictly prove that there are such entities as qualities, i.e. the universals represented by adjectives and substantives, whereas we can prove that there must be relations, i.e. the sort of universals generally represented by verbs and prepositions.

Let us take in illustration the universal whiteness. If we believe that there is such a universal, we shall say that things are white because they have the quality of whiteness. This view, however, was strenuously denied by Berkeley and Hume, who have been followed in this by later empiricists. The form which their denial took was to deny that there are such things as 'abstract ideas '. When we want to think of whiteness, they said, we form an image of some particular white thing, and reason concerning this particular, taking care not to deduce anything concerning it which we cannot see to be equally true of any other white thing. As an account of our actual mental processes, this is no doubt largely true. In geometry, for example, when we wish to prove something about all triangles, we draw a particular triangle and reason about it, taking care not to use any characteristic which it does not share with other triangles. The beginner, in order to avoid error, often finds it useful to draw several triangles, as unlike each other as possible, in order to make sure that his reasoning is equally applicable to all of them.

But a difficulty emerges as soon as we ask ourselves how we know that a thing is white or a triangle. If we wish to avoid the universals whiteness and triangularity, we shall choose some particular patch of white or some particular triangle, and say that anything is white or a triangle if it has the right sort of resemblance to our chosen particular. But then the resemblance required will have to be a universal. Since there are many white things, the resemblance must hold between many pairs of particular white things; and this is the characteristic of a universal. It will be useless to say that there is a different resemblance for each pair, for then we shall have to say that these resemblances resemble each other, and thus at last we shall be forced to admit resemblance as a universal. The relation of resemblance, therefore, must be a true universal. And having been forced to admit this universal, we find that it is no longer worth while to invent difficult and unplausible theories to avoid the admission of such universals as whiteness and triangularity. ...

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'. ... In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness. The connected ambiguity in the word 'idea', which we noted at the same time, also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'.

Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.

Consider that when you think about triangularity, as you might when proving a geometrical theorem, it is necessarily perfect triangularity that you are contemplating, not some mere approximation of it. Triangularity as your intellect grasps it is entirely determinate or exact; for example, what you grasp is the notion of a closed plane figure with three perfectly straight sides, rather than that of something which may or may not have straight sides or which may or may not be closed. Of course, your mental image of a triangle might not be exact, but rather indeterminate and fuzzy. But to grasp something with the intellect is not the same as to form a mental image of it. For any mental image of a triangle is necessarily going to be of an isosceles triangle specifically, or of a scalene one, or an equilateral one; but the concept of triangularity that your intellect grasps applies to all triangles alike. Any mental image of a triangle is going to have certain features, such as a particular color, that are no part of the concept of triangularity in general. A mental image is something private and subjective, while the concept of triangularity is objective and grasped by many minds at once.

Instead of questioning the existence of abstract object, perhaps we should apply our skepticism the existence of an exterior physical universe.

Bowdlerising his argument, it simply is not the case that the grey of a cloud and the grey of this laptop have something in common - apart from our use of the word "grey".

Bowdlerising his argument, it simply is not the case that the grey of a cloud and the grey of this laptop have something in common - apart from our use of the word "grey". Or if you prefer, abstract objects do not exist.

Abstract objects don't "exist" in any particular mind. Pi is an abstract object. It's not a resident of my mind in the way my grocery list is. I can't be wrong about my grocery list. I can be wrong about pi. It's that sort of thing.

Abstract objects, last I checked, have been, at the very least, more closely associated with the mind than the physical world.

Also, for the moment, ignore the notion of abstract(ion) and give me an example, if there is one, of an object that's exclusively mind, having no connection at all with the physical world.

They aren't mental objects.

I don't know.

Earl Gray?

But a difficulty emerges as soon as we ask ourselves how we know that a thing is white or a triangle. If we wish to avoid the universals whiteness and triangularity, we shall choose some particular patch of white or some particular triangle, and say that anything is white or a triangle if it has the right sort of resemblance to our chosen particular. But then the resemblance required will have to be a universal. Since there are many white things, the resemblance must hold between many pairs of particular white things; and this is the characteristic of a universal. It will be useless to say that there is a different resemblance for each pair, for then we shall have to say that these resemblances resemble each other, and thus at last we shall be forced to admit resemblance as a universal. The relation of resemblance, therefore, must be a true universal. And having been forced to admit this universal, we find that it is no longer worth while to invent difficult and unplausible theories to avoid the admission of such universals as whiteness and triangularity. ...

Or if you prefer, abstract objects do not exist.

Brief defense of universals, bolds added.

Russell, World of Universals:But universals do not exist in this sense; we shall say that they subsist

Can you give me an example, one will do, of a pure abstract object and by that I mean an (abstract) object that has no links whatsoever with the physical world? It should exist only in the mind is what I'm saying.

OK, if we require “exist” to apply to only things in space/time, then universals don’t exist but they subsist. But if we make this requirement of "exist", then it seems “is” and “was” are fundamentally different.

the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. If we ask 'Where and when does this relation exist?' the answer must be 'Nowhere and nowhen'. There is no place or time where we can find the relation 'north of'. It does not exist in Edinburgh any more than in London, for it relates the two and is neutral as between them. Nor can we say that it exists at any particular time. Now everything that can be apprehended by the senses or by introspection exists at some particular time. Hence the relation 'north of' is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something.

The luminiferous aether is an abstract object that the universe fails to instantiate.

Or if you prefer, abstract objects do not exist.

That's just not true, though; they each reflect light at wavelengths closer to each other than objects of other colours do compared to them, and consequently they look more similar to each other in terms of colour than objects of other colours do compared to them. The first is a material condition of the second and the second is the reason we refer to both as being grey, for our use of the word "grey".

Without the idea of two, we cannot apply the idea of two to a pair of apples. Example, I define “xyz” as the set of all xyz things. Not a very useful definition.

An analogy: Imagine indirect experience as watching a baseball game on TV, as opposed to being in the park. We don’t directly experience the tree; our senses play the role of TV.

Apple is a universal. A particular apple is an instantiation of the universal called “apple”.

Without the idea of two, we cannot apply the idea of two to a pair of apples. Example, I define “xyz” as the set of all xyz things. Not a very useful definition.

Moreover, the set of all existing two things is constantly changing. If I eat one of the two apples, then the “set of all existing two things” has changed. If two atoms are crushed out of existence in some neutron star in another galaxy, the “set of all existing two things” has changed.

Ideas exist in the “mindscape.” Physical cats exist in the physical world.

Experience is concrete. I physically experience rough brown patches and smooth green patches, which lead me to mentally experience a universal, i.e., the idea of a tree.

From where did we get the idea of two if not by first observing more than one thing?

How did you come to experience the universal by observing just one pattern (a particular) of rough brown patches and smooth green patches?

My view is that ideas already exist in the mindscape, just as trees exist in the landscape. Seeing a pair of apples may awaken our mind to the idea of two, but the idea already exists. Any being which lacks the mental capacity will never perceive the idea "two." Imagine an earthworm, for instance, crawls over two pebbles. I doubt the idea of two ever enters what mind it has.

Can you have an idea of "tree" without having first perceived more than one tree?

What is the difference between the idea of the universal tree vs the particular tree?

What does the idea of two look like in the mind independent of the scribble, "2" or "two" and independent of the observation of two particulars?

How do you know that you are holding the idea of 2 in your mind independent of these forms

they view all objects as individuals

Why would I need to hypothesize that inhabitants who use universals can perceive some Platonic realm, when I simply can appeal to our make up that favors detecting commonalities and creating language of universals vs detecting differences and name individuals?

So light with a wavelength of 650nm is the same colour as light with a wavelength of 651nm because they're very similar wavelengths?

The literal meaning of 'subsist' is nothing like what he's trying to convey here.

That's just not true, though; they each reflect light at wavelengths closer to each other than objects of other colours do compared to them, and consequently they look more similar to each other in terms of colour than objects of other colours do compared to them. The first is a material condition of the second and the second is the reason we refer to both as being grey, for our use of the word "grey".

“Object” should not be taken too literally; think “abstract entity.”)

How about the following? "Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), something that waves should not be able to do."
The luminiferous aether is an abstract object that the universe fails to instantiate.

Just keep in mind that if you find that your analysis of abstract objects contains the very thing you're trying to analyze (such as 'things we do', which is a set, and therefore an abstract object), you may take that as a signal that what you're dealing with is more primal than you previously may have thought.

But at least at this point, you've recognized that they exist and all that's left is to join the ongoing debate about how to understand them.

thanks, hadn't noticed that second usage. Careless on my part. Hope it doesn't detract from the main point.

a. to exist as a concept or relation rather than a fact
b. to be conceivable

I would argue, are numbers abstract

There might be an error even here. Perhaps at least some of what you call "abstract objects" are things we do, not things we find.

Those with a background in philosophy may recognise what I am suggesting as deriving from philosophy of language. Instead of looking for the meaning of the terms we use, stand back and look at how they are used.

'things we do', which is a set, and therefore an abstract object

The aether is a medium for waves, both these concepts have links to the physical (water waves/ripples).

there's a controversy with regard to whether math is invented or discovered.

I see problems with defining abstract objects in terms of sets

I would argue, are numbers abstract — Janus


That is the prevailing view in philosophy of math.

But 'NO!' the physicalist will cry 'the view that the physical is primary is the more plausible'. Yet plausibility is not a precise measure and a sense of it is gained only by comparing many cases, and in this connection we have only the one case to consider. So, it seems that our sense of plausibility here is merely a reflection of conditioned habit and the dominant paradigms of our social milieu.

If you're a physicalist, you probably accept that in some sense the universe is aware of itself, so I don't know if it's more plausible. It's just the starting point our culture embraces.

Does this idea of the universe being aware of itself mean that the whole universe is aware of it wholeness, or just that some parts are aware of parts? Of course the latter is uncontroversial, as animals and humans seem to show various degrees of self-awareness.

I think of awareness as usually being like a flashlight in a dark room. I just meant that physicalism, to the extent that it's monistic, has to accept that the universe has awakened to itself. That's what we are.

Processes are not a problem. Swimming, runnning, etc. are universals.

True, the language we develop has indications of what exists but if our interest is ontology, language is subordinate to reality. In the middle ages, language included much talk of "witches" but that didn't mean witches really exist.

I think that it is undeniable that without awareness the universe would be as good as nothing. It might be said to exist in some sense, but it would be an entirely blind, deaf, dumb and senseless existence, whatever we might dimly be able to imagine that could be.

an abstract object does not exist in space and time.

A geometry teacher is not trying to teach about what exists in his/her mind but the triangle "out there." Question: suppose I say triangles exist in my mind and they have four sides. How could anyone dispute what I say? Sure, triangles in another person's mind might have three sides but so what? Triangles in someone else's mind might have five sides. Clearly a definition of triangle is needed. Would you agree that definitions exist in the external world not only in our minds?

Ideas exist in the “mindscape.” Physical cats exist in the physical world.

A lot of Cartesian dualism here— mind/body, subject/object, mental/physical, inside/outside.

Cartesian anxiety refers to the notion that, since René Descartes posited his influential form of body-mind dualism, Western civilization has suffered from a longing for ontological certainty, or feeling that scientific methods, and especially the study of the world as a thing separate from ourselves, should be able to lead us to a firm and unchanging knowledge of ourselves and the world around us. The term is named after Descartes because of his well-known emphasis on "mind" as different from "body", "self" as different from "other".

I think there’s something to that, yes. I don’t know who Bernstein is, but I bet he read Heidegger.

Bernstein diagnosed a serious issue that affects much of modern philosophy as it oscillates unendingly between two untenable positions; on the one hand, the dogmatic search for absolute truths, and on the other, the conviction that “anything goes” when it comes to the justification of our most cherished beliefs and ideas. According to Bernstein, what underlies this predicament is a deep longing for certainty, the urge “to find some fixed point, some stable rock upon which we can secure our lives against the vicissitudes that constantly threaten us.”[10]

This is what he calls the Cartesian anxiety, a mostly unacknowledged existential fear that seems to lead us ineluctably to a grand Either/Or: “Either there is some support for our being, a fixed foundation for our knowledge, or we cannot escape the forces of darkness that envelop us with madness, with intellectual and moral chaos”.

Although in philosophy this Cartesian anxiety mostly shows up in the discussion of epistemological issues, Bernstein is pointing to something much deeper and universal with this notion, something that permeates almost every aspect of life and has serious ethical and political consequences.