PHIL2109 Contemporary Metaphysics
Week 13: Abstracta

A distinction

  1. There seems to be a fundamental difference between the kinds of things in the first list below and the kinds of things in the second:
  2. Things of the kind in the first list are commonly called concrete things (or concreta); things of the kind in the second list are commonly called abstract things (or abstracta).

Drawing the distinction

  1. Assuming that there is a difference, what is it? How should we draw this distinction?
  2. One natural way is as follows:
  3. Here are some issues for this criterion:
  4. Here is another way to draw the distinction:
  5. Here are some issues for this criterion:
  6. Perhaps there are two senses of the words 'concrete' and 'abstract', so that there are two distinctions to draw. And perhaps the first proposal correctly draws the first distinction, and the second proposal correctly draws the second.
  7. But here is a concern about both. How does either allow that we can know that there are such things as numbers, sets, and so on, if they are abstract things, and if abstract things are either non-spatiotemporal or non-causal? How can we even be justified in believing that there are such things? Relatedly, how can we know or justifably believe anything about them?

    Some possible replies:


  1. Numbers are often pointed to as prototypical abstract things. This assumes that there even are such things as numbers. Are there?
  2. Here is an argument that there are:
    1. It is true that some numbers are even.
    2. If it is true that some numbers are even then some numbers are even.
    3. If some numbers are even then there are numbers that are even.
    4. If there are numbers that are even then there are numbers.
    5. Therefore, there are numbers.

    If the first premise here is contentious, then we could start with the premise that it is true that 2 + 2 = 4, which has to be the least contentious premise we could come up with.

  3. Here is a related argument:
    1. It is true that F = Gm1m2/r2. (Newton's law of gravitation)
    2. This would not be true if there were not numbers (or functions).
    3. So there are numbers (and functions).

    But need physical laws be true? Even if they are, why do there have to be numbers and functions for them to be true? Perhaps they can be formulated without any appeal to numbers or functions.

  4. Ok, suppose there are numbers. Might they not be concrete things?
  5. Here is a bad argument that they are not:
    1. There are infinitely many numbers.
    2. There are not infinitely many concrete things.
    3. Therefore, numbers are not concrete things.
  6. Here is a reason to think that they might be concrete. We can account for the truths of mathematics by taking numbers to be all manners of things. We can take them to be sets in either of the following two ways:

    We could even take them to be various concrete things, as long as they stand in the right relations to each other. So perhaps numbers are concrete things.


  1. If it is too contentious to point to numbers as being abstract things, perhaps it is less contentious to point to sets.
  2. But, granting that there are such things as sets, might they be concrete things? Perhaps a set of concrete things is itself a concrete thing, located in space and time exactly where its members are located in space and time, and/or having exactly the causal powers of its members?
  3. Or perhaps there are no such things as sets, because talking about 'sets' is just a convenient way of talking about the things which we call 'members' of the set? E.g. talking about the set {Sydney, London} is just a convenient way of talking about Sydney and London. E.g. saying the fist below is just a convenient way of saying the second or third:

Possible worlds

  1. Here is an argument that there are abstract things: There are possible worlds; possible worlds are abstract things; so there are abstract things.
  2. We have seen one argument in favour of the first premise (due to Lewis): Things might have been various ways; so there are various ways things might have been; so there are various possible worlds.
  3. Are these possible worlds concrete things? According to Lewis, yes - they are things of the same kind as our actual world, and just as concrete. But that is to claim that ways things might have been are the same kind of thing as our actual world. To be consistent, Lewis should say that the way things are is the same kind of thing as our actual world, but that seems to be a category mistake. So there is no good reason here to think that possible worlds are concrete things.
  4. Note also that, according to Lewis, possible worlds other than our own are completely spatiotemporally and causally isolated from us. Is that a reason to think that they are not concrete? Or is that a reason to doubt these two ways of drawing the distinction?
  5. Let's forget Lewis's extra claims and focus on ways things might have been - are they concrete things? According to Stalnaker they are properties. But whether or not properties are concrete is contentious.

Laws of nature

  1. There seem to be such things as laws of nature. Let's suppose that this is one of them: Heavy objects sink in water.
  2. What kind of thing are we saying here? One idea is this: we are expressing an exceptionless regularity: every case in which a heavy object is placed in water is a case in which it sinks.

    But laws of nature support counterfactuals: If I were to place this heavy object in water then it would sink; Why? Because heavy objects sink in water. The present proposal seems to not account for this.

  3. Here is another idea: we are saying that a certain relation obtains between two properties - the property of being a heavy object, and the property of sinking in water. Which relation? Some kind of necessitation relation - not logical necessitation but natural necessitation.

    If this is right, and if the law is true, then there are properties. And if properties are abstract, then there are abstract things.

    But Lowe raises a question for this approach to laws: Why should the particulars obey the law, if it is a relation between properties?


  1. We have arrived at the following position: whether or not there are abstract things comes down to whether or not properties are abstract things. Are they?
  2. Here is a possibility: Properties are concrete; there are no uninstantiated properties; a property is located in the location and time of the particulars that instantiate it (so many have a scattered spatiotemporal location).
  3. But here is an objection to this idea. To explain sameness between two particulars, advocates of properties need to claim that a property is wholly located in each of its instances. But that means that if blueness is wholly located where pen A is, and also wholly located where pen B is, then pen A is wholly located where pen B is, which is absurd.
  4. So we left wondering whether or not there really are any things which are not concrete.