On Philosophy

March 22, 2007

Two Kinds Of Existence

Filed under: Metaphysics — Peter @ 12:00 am

What do we mean when we say that something exists? The obvious answer is that it exists physically, that it is part of the physical world and that we can interact with it. But what about numbers, do they exist? It seems sensible to say that they don’t exist in this way, but given that we talk about them as if they did exist in mathematics it seems possible to defend the idea that they must exist, in order to make sense of talk about mathematics. Such a proposal would expand the universe to include abstract entities in addition to the familiar physical entities. But that is not a satisfactory solution either; it opens the door to the existence of all sorts of things, like an infinite number of possible objects. And when you admit that possible objects exist you have become confused about what existence means, or at least a very poor grasp of Occam’s razor. Possible objects are only possible because they don’t exist, because they aren’t part of the physical universe. And moreover a universe that contains all possible objects is simply metaphysically overburdened; possible objects bear no explanatory burden except to explain certain forms of discourse, and it would be a simpler conception to explain such discourse in terms of false ideas about what the universe contains.

So let us go back to the argument that the numbers must exist in some fashion. Certainly we say many things, supposedly true, about the numbers as if they existed. For example we say that “5 is a number”. In addition we believe that there is a fact of the matter about certain statements, such as “there are infinitely many twin primes” that are either determinately true or determinately false depending on the structure of the natural numbers, even though we don’t know yet whether they are true or false. Which implies that somehow the natural numbers are “out there”, beyond our conception of them, making these statements true or false.

But this argument turns, I think, on an equivocation between two senses of existence. Existence, in one sense, says only that something is part of the physical universe, nothing more. In another sense existence means something that can be reasoned about or talked about, logical or linguistic existence. The numbers, I would say, have a logical or linguistic existence. And this doesn’t mean that they are extra non-physical entities that are also part of the universe. We don’t interact with “the numbers” or have access to them in any special way. Rather these logical entities are created by our discourse and its rules. For example, I could define space as being divided into cubic regions one meter on each side. I could talk about the contents of these regions, their arrangement, ect. So in a sense these regions “exist”. And there is a fact of the matter about which region any given object is in, even if you don’t know which one that is. But that doesn’t mean that these arbitrary divisions of space are real, they simply reflect a way of talking about space.

And so it seems reasonable to say that numbers, and possible objects, have a logical or linguistic existence as well, since they are no more independent of the rules we have made about them then our arbitrary divisions of space were. They are entities that we think of, talk about, and reason about as if they exist. They are defined by certain rules, and the rules may imply certain facts that we aren’t aware of (because we can’t immediately understand all the logical consequences of those rules). And if they are useful they probably seem to reflect certain facts about the physical world. For example, the numbers seem to reflect certain facts about the cardinality of sets of physical objects and certain operations on those objects.

Of course none of this disproves the idea that these abstract objects exist as a kind of extension to the physical world. But it does show that we don’t need to add them to the world in this manner in order to explain how we can reason about them, and how there can seem to be facts about them independent of our conception of them. And thus, since it explains all the same facts, we should accept that simpler hypothesis that they exist only linguistically.



  1. Dear Peter,

    You pose an interesting concept, and I am more than happy to convey my thoughts to you about this subjective argument.
    Even though I only have an eighth grade education (I failed math), I am currently writing a book called:

    The Theory of Numerical Sequential-Cancellation Renormalizations

    The subject matter of my work is not important at this time, however, I do believe I at least have some idea in regards to your interesting concept: NUMBERS…Two Kinds Of Existence: Logical/Linguistic?

    First, I would like to form a basis of understanding as to what numbers in nature are initially. Ands since I have a fascination for sequences, (such as Fibonacci sequence) and frequencies and the like, I find that numbers, in nature, follow one golden-lovely, indisputable rule (as it were): They describe the path of least resistance where all matter is concerned!!!


    That is the logical basis; but what about the matter of “linguistics”? I believe that math, like written music per-say, is the language that we use to describe what ever it is you wish to try to convey and express in a mathematical detail.

    The question is: Does nature know how to do math? Or is math merely a description of matter blindly traveling through space-time the easy way?

    Your response will be greatly appreciated, Peter. Your article is very intriguing!

    —Robert Darrell

    Comment by Robert darrell — March 22, 2007 @ 4:39 am

  2. I would say that mathematical laws are a description of events, not a cause of them, for reasons of simplicity if nothing else, because to suppose that they are a cause of events would be to grant them the same kind of existence as physical objects, which I am obviously opposed to. (The physical domain being defined by its causal closure, anything that can be the cause of a physical event is itself physical.)

    Comment by Peter — March 22, 2007 @ 5:00 pm

  3. Dear Peter,

    It is an honor and a pleasure to communicate with such a very intelligent person as yourself. You are amazing!

    I never thought to ask myself to consider the possibility that math could possibly be an entity in itself. It is to my dissatisfaction that I never thought of it. It was not until after I read your article about the subject that I became aware of it.

    Of course, I quite agree with you when you stated that “…anything that can be the cause of a physical event is itself physical.”

    You put that beautifully.

    Peter, it took me almost ten years to solve certain mathematical enigmas, regarding my work, even though my acquaintances said it could not be done—What I am saying is, is that it is not in my nature to let this matter go. I spent some time this morning meditating over this subject of math and the results end up the same: I cant seem to correlate a mathematical idea as a physical entity in itself. Not yet! I have only just begun to analyze the concept— And I am really enjoying this.

    If I happen to think of something fascinating regarding this subject, I promise you will be the only person I know to talk about it with!—If you think of something, please elaborate!

    Anticipating Your Reply,

    Your Friend, Robert

    Comment by Robert Darrell — March 23, 2007 @ 8:29 am

  4. Post Script: By the way…You may visit me at my favorite website…


    Comment by Robert Darrell — March 23, 2007 @ 12:35 pm

  5. If you are interested in more about whether abstracy entities (of which numbers are one kind), in this case properties, are better thought of as descriptions of regularities in nature or behind those regularities I have a post titled “What Are Properties?” on the subject which I will put up in three days (the 26th).

    Comment by Peter — March 23, 2007 @ 8:28 pm

  6. Exellent!
    Looking forward to “What Are Properties”

    Comment by Robert Darrell — March 23, 2007 @ 11:49 pm

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