Recap: Alyssa Ney's Metaphysics: Chapter 2 - Abstract Entities

2.1 - More Than a Material World?

• Now that we have Quine's method of regimentation for determining our ontological commitments, we can apply it to abstract entities like numbers to see whether we are committed to their existence.
• As a reminder, that method is the following:
• Step 1: Decide which sentences you take to be true.
• Step 2: Regiment the sentences by symbolizing them in the language of first-order predicate logic.
• Step 3: Commit yourself to all and only those entities needed to stand in as the values of the bound variables in order to make the sentences true.
• However, before we apply the method, we should get more clear on what it means to be abstract. How do we separate the abstract entities from the concrete ones?

2.2 - The Abstract/Concrete Distinction

• Concrete objects are those most familiar to us, like tables, chairs, lakes, mountains, stars, planets, people, and animals.
• These objects have causal properties, i.e., they stand in causal relationships. For example, the carpenter causes the table to exist, and the table causes me to think of the table when I see it.
• Question: Is the number 2 causing me to think about 2 when I do? If yes, then doesn't that mean 2 is causal? We could apply this to fictional characters. Is Pegasus causing me to think about Pegasus? No, but fictional stories (or philosophical discussions) about Pegasus cause me to think about Pegasus. But if we apply this to numbers, then does that mean every mathematical equation is like a fictional story? But how could that be when math accurately describes reality?
• We typically think of the following as abstract objects: 
• Numbers
• Does the number 2 exist? Do sets exist? Are imaginary numbers any less real than real numbers?
• Shapes
• Do squares or triangles exist?
• Properties
• Properties are those things we use to describe objects, such as shape, size, mass, velocity, luminosity, material composition, ductility, temporal location, spatial location, and so on.
• Do properties exist, or do real objects simply have properties?
• In philosophy of mind you have the view aspect dualism, which says that mental properties cannot be reduced to physical properties. Put another way, first person and third person properties are entirely separate categories of properties.
• Third-person properties include those mentioned above.
• First-person properties include, among others, aboutness (aka intentionality) and first-person private experience.
• Propositions
• Sentences express propositions, but they aren't the same thing.
• The following two sentences are different (pg 56):
• Everyone is in a good mood today.
• Tutti sono di buon umore oggi.
• But they both express the same proposition.
• Propositions are abstract. (What exactly they are will be addressed in Ch 10.)
• But words are thought of as concrete, as they are composed of concrete symbols (symbols we create) or sounds (sounds cause us to hear). Sentences, which are made up of words, are concrete too.
• So sentences give us a tangible, concrete access to the intangible, abstract propositions they represent.
• Virtues and vices
• Does bravery exist, or selfishness? Do acts of bravery or acts of selfishness exist?
• Fictional characters
• We've already discussed one fictional character, which is Pegasus. In that discussion I concluded that I am not committed to the existence of Pegasus, but I am committed to the existence of thoughts about Pegasus.
• Are thoughts concrete or abstract? I take it that thoughts are concrete because they are causal objects. My brain plays an important role in causing my thoughts (if I had no brain I would have no thoughts). Plus my thoughts cause things (sad thoughts cause me to cry and funny thoughts cause me to laugh). And my thoughts have temporal locations; some thoughts come before or after other things. The number 2 does not come before or after anything.
• While tables seem clearly concrete, and numbers seem clearly abstract, there are some tricky cases (pg 57). Are fundamental particles concrete? What about space itself? Or time itself? Or colors? Or the colors of individual objects? Or events?
• Abstraction is the process of abstracting out some property from an object. So from the concrete object of a table we can abstract out the rectangular shape and the brown color of the table. So from the concrete object of the table we can find the abstract object of its shape using abstraction. So abstraction helps us identify the properties of objects, and properties are themselves abstract objects.
• Question: Are all abstract objects properties, or bundles of properties? Or more fundamentally, are all objects bundles of properties?
• Can we abstract out abstract objects from abstract objects? It seems we can: the number 2 has certain properties, like being greater than 1 and less than 3. But I can ignore all those properties and focus on one: being divisible by 2 without a remainder. Divisible-by-2-ness is an abstract object, a property, that was abstracted out from another abstract object, the number 2. All even numbers have this property.
• Question: Are we committed to all concrete objects?
• Exercise 2.1 asks us to label a list of objects as abstract or concrete. One of them is God. Normally I would say God is a concrete object; God creates the world and thus stands in causal relations. However, while that's part of the definition of God, I don't actually believe in God. So God doesn't really stand in any causal relations. So whether God is causal or not depends on whether he exists. Non-existent objects are not causal (you cannot create a non-existent object and nor can a non-existent object affect anything). However, thoughts about non-existent objects are causal and do exist (assuming thoughts are concrete objects like I've argued). So thoughts about God are always causal while God per se is only causal if he exists.
• What causes my thoughts about God if not God? Here David Hume helps us. (See E 2.5, SBN 19 of An Enquiry concerning Human Understanding, https://davidhume.org/texts/e/2). David Hume says God is an abstraction of an idealized person. As Hume explains, or at least suggests at, our imagination can combine properties in impossible ways. So exercising our imagination allows us to generate thoughts of impossible combinations of properties and thus of impossible objects. Then people write down what they imagined into stories. These stories of God (and other fictional characters) then cause other people to think about God (or those characters). Impossible objects are made up of impossible properties (or of an impossible combination of possible properties). So impossible objects do not cause us to think of them; our imaginations do.
• So to answer the question above, we are not committed to causal objects that aren't real... obviously. But we are committed to those causal objects that are real... obviously. God would be a causal object if he were real. But he's not, so he's not. (It's okay to say "God is a concrete object," just like it's okay to say "Pegasus is a white stallion with wings." We are speaking in terms of identity, not of predicate. That is, we're not saying, "God exists as a concrete being," we are saying, "Part of what it means to be God is to be a concrete being.")

2.3 - Universals and the One Over Many Argument

• Now we are focusing a bit on properties. Do properties exist? Round tables exist. But does roundness exist?
• A universal is an entity that is found in many places at once; it's instantiated repeatedly at the same time. A particular is an object that is not universal.
• Random thought: We are not committed to any object that can be reduced to another object. For example, we are not committed to the existence of darkness, because we can reduce darkness to the absence of light. Cold likewise is the absence of heat. (Christians often say badness is the absence of perfection. But pain is certainly bad and certainly not an absence. But if pain is real, then the pain of being cold is real. And so cold is real. Except the pain of cold can be reduced to the pain of a lack of heat. So the analogy between darkness / cold and badness fails; the badness of pain cannot be reduced to a lack of pleasure or a lack of God's will or something.) The parsimonious view then is the view that reduces all that can be reduced as far as possible so we end up with the fewest fundamental objects.
• Wait... do absences themselves reduce to an absence? Nothing can reduce to itself. So absences are fully reduced abstract objects. An absence of light is the same abstract object as darkness, and darkness is the same abstract object as a lack of light. So a lack and an absence are the same thing.
• Holes are absences. Saying, "there are holes in this Swiss cheese" is the same as saying "There are absences of cheese in this Swiss cheese" 
• Holes being absences makes perfect sense because if you were to fill the Swiss cheese with a maximal number of holes, there would be a total lack of Swiss cheese. (Imagine the cheese being filled with more and more holes until it was gone.)
• Likewise, the presence of light is the same thing as light. To have the presence of light just is to have light. To have an absence of light just is to have no light.
• So we are not committed to presences or absences. So we are not committed to holes. Though, holes are abstract objects, because we can abstract out hole-ness. So there are at least some abstract objects that we are not committed to. (We already knew that. Fictional, non-existent, and impossible objects are all abstract objects. Sets and species are also abstract objects that we seem to not be committed to.)
• Question: Are properties the only universal?
• Relations are universals. Being next to something is a relation, and the relation of being next to something is instantiated all over the place.
• My night stand is next to my bed. So my night stand instantiates the property of being next to such and such bed.
• Huh, so it seems like relations are properties.
• Okay, end of random thoughts.
• Instantiation is the relation between a property and an object that has that property. Basically, it's a fancy word for 'has'. The number 2 has the property of evenness, so 2 instantiates evenness. God instantiates perfection, the table instantiates roundness and being colored brown and being in a next to relation with another table, and so on.
• Abstract objects can instantiate properties, not just concrete objects. And since we know there are abstract objects that don't exist, this means instantiating properties does not entail existence.
• Propositions can instantiate the property of being true or false. So cool!
• One object can have a seemingly endless number of properties. Take the human body for example. For each organ, you instantiate the property of having that organ. For each dust mite on you, each bacterium, each molecule that makes you up, you instantiate being made out of those things or having those things in you or on you. Because of this, we often distill objects to a few key properties when talking about them to simplify things. While a table will technically instantiate the property of having this kind of scratch on this part of the table and this bit of rust on this part of the under-framing, these super detailed properties are not relevant to us most of the time, so we can happily ignore those.
• Okay, end of random thoughts again.
• Aristotle gives the example of being human as a universal, while Callias is a particular, in this case a particular human.
• Plato believed in special universals called Forms. The Forms are real entities in Plato's ontology. They are perfections of certain kinds. Beauty, Justice, and the Good are all examples of Forms.
• Random thought: Ugh, sorry. It's me again. I suddenly found myself really liking the A theory of time. I know it's not a popular theory in some circles. But it eliminates all past and future events and objects. They don't exist! That's amazing! That's so much more parsimonious than the alternative. That parsimony is beautiful. This fits nicely with my intuition that when things are destroyed, they go out of existence. When events are over, they are no longer real. The past feels done and over with.
• I really see the appeal of nominalism. It would be really nice to say that all abstract entities do not exist (put another way, we are not committed to the existence of any abstract objects). That feels like a beautifully simple worldview.
• Josh Rasmussen says separate the clear from the unclear. It's clear to me that thoughts exist. It's clear to me that events exist (for example, the event of me having had a thought). But is it clear that properties exist, or numbers, or any other abstract objects? It's clear that thoughts about those things exist. I'm happy with words and sentences existing. But do propositions exist? If truth is an abstract object, what would it mean to say truth isn't real?
• For now, I will believe that some abstract entities do not exist and remain agnostic about the others until I get a clearer picture of things.
• It seems like a more basic and obvious method than Quine's is simply to say "Numbers exist" and then see how you feel. I feel mixed when I hear that one. "Unicorns exist." I feel like that's obviously false. "Thoughts exist." That's obviously true. I guess this is Step 1 of Quine's method. But then regimentation isn't doing much beyond restating our intuitions.
• Realism = the view that abstract objects / properties / universals do exist as mind-independent objects.
• Nominalism = the view that abstract objects do not exist.
• Platonism = a specific kind of realism, though sometimes is just another word for realism.
• Conceptualism = abstract objects do exist, but they are mind-dependent.
• One Over Many = an argument in favor of realism.
• 1 - There are red houses, red roses, and red sunsets.
• 2 - Therefore there is this one, redness, that runs through the many.

2.4 - Applying Quine's Method

• The One Over Many argument, while a classic argument, is invalid. 2 does not follow from 1.
• There are red houses, red roses, and red sunsets = ∃x∃y∃z((Rx∧Hx) ∧ (Ry∧'Ry) ∧ (Rz∧Sz))
• There is an x, y, and z such that x is a house and red, y is a rose and red, and z is a sunset and red.
• This commits you to red houses, red roses, and red sunsets, but not to redness itself. 
• David Armstrong says we are committed to the existence of redness because only the existence of redness can explain how it is that red houses, red roses, and red sunsets could all be different objects and yet be meaningfully similar.
• It is true that red houses, red roses, and red sunsets share in a property. But what is it that makes this true if not for the existence of the property of redness?
• This is similar to the question: It is true that 2 + 2 = 4. But what makes this equation true unless 2 and 4 are real (and the relations of 'plus' and 'equals')?
• Truthmaker theory = the theory that truths have truthmakers; propositions are truth-bearers, and propositions are made true, or false, by something (such as by facts, also known as states of affairs).
• Armstrong says truthmakers are states of affairs.
• Truthmaker maximalism = the theory that all truths have truthmakers. This theory is very controversial, and is part of a broader debate surrounding the correspondence theory of truth.

Note: Second-order predicate logic does allow one to quantify over properties, and thus allows the One Over Many argument to succeed. But Quine has suspicions about second-order predicate logic as a system. One, it lacks completeness unlike first-order predicate logic. Two, the quantification over properties is built into second-order predicate logic, implying that it's a matter of logical necessity that properties exist. But Quine says whether something exists should be a matter of investigation. Logic tells us what follows from what, not what exists. First-order predicate logic does not have this problem; it does not assume the existence of anything.

2.5 - Nominalism and Other Options

• Truthmaker theory presents an important challenge to nominalism. Remember that explanatory power is an important theoretical virtue. Often, in order to obtain an explanation, you must posit an entity. Theists believe that if you want to explain the universe then you'll need to posit God. So the big challenge to nominalism is how nominalism can secure explanatory power while positing so few entities. Realism is an attractive position because of the explanatory power it promises.  
• Predicate nominalism = This view says statements like "The house is red" can be true even though redness does not exist. Nothing is needed to explain the truth of the proposition. This is the view Quine defended.
• I don't like the idea of burying one's head in the sand in this way. I want to have both a parsimonious ontology while still being able to explain why truths are true. Though this view looks nice for those who don't like the correspondence theory of truth / truthmaker theory. But I like correspondence.
• Class nominalism = The theory that properties are sets of all the individual objects that instantiate that property. A property is not a universal on this theory, but a particular class.
• The problem of co-extension looks bad. Trilaterality is not exactly the same as triangularity.
• Goes against my intuitions. Being divisible by 2 is a property that some numbers instantiate. Objects have properties. But on this view objects don't have properties, but belong to properties as members of a set.
• Trope theory = Like class nominalism, this theory says there are no universals. Properties are abstract particulars called tropes. A red house, red rose, and red sunset actually do not share the same redness on this theory. Rather, each has their own particular red trope. All objects are bundles of tropes. This theory is associated with D.C. Williams.
• My first impression of trope theory is not good. The whole point of abstraction is to remove particularity, right? And, if two objects have identical colors (say, hexadecimal #808080, or perfect gray), then you can’t say that they are different instances of color. It's the exact same color. The only way to differentiate the colors would be to connect the colors to the objects themselves; the color from this object and the color from that object. But that would mean the identical colors are particulars in virtue of their relations to other properties of the host object like space, time, age, etc. That means we cannot abstract anything out. The whole point of abstraction is to isolate properties. But it looks to me like trope theory cannot isolate properties, at least not when two objects share an identical property. But when we write fictional characters, what are we doing if not isolating properties and combining them in interesting ways?
• I do like the idea that all objects are bundles of properties, though if I'm not mistaken this would commit me to properties because some objects certainly exist and they would reduce to their bundle of properties. 

2.6 - Mathematical objects

• To decide whether numbers exist, we can apply the same moves as before with regimentation, but we will run into the same problems. The nominalist shows by first-order predicate logic that they are not committed to numbers and the realist shows how the non-existence of numbers renders mathematical expressions false because there are no truthmakers to make mathematical statements true. Or, the non-existence of numbers makes the truth of mathematical expressions mysterious.
• However, there is another move: the indispensability argument.
• This argument is associated with Quine and Hilary Putnam. It goes like this:
• 1 - We ought to have ontological commitment to all that is indispensable to science.
• 2 - Numbers are indispensable to science.
• 3 - Therefore we ought to have ontological commitment to numbers.
• However, science makes widespread use of idealizations, which are false assumptions that simplify things in a very useful way, making calculations much easier while maintaining a useful degree of accuracy.
• In thermodynamics, the gas law is simplified to assume that particles do not interact when really they do. In physics, surfaces are assumed to be frictionless when really they are not.
• And in fluid dynamics, fluids are assumed to be uniform and not made up of discrete atoms when really they are (https://youtu.be/5M6zXKNFgIo?t=1667).
• Penelope Maddy notes that we must allow for the distinction between the parts of science that are true and the parts that are useful.
• So numbers could just be useful fictions like idealizations.

As always there's more to the chapter but I will leave it there.