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I am just going to give an example of what I mean using Skolem's Paradox. I DO NOT want to get into Skolem's Paradox itself or its "resolution."

Skolem's showed that countability is relative in models of first-order formulations of ZFC (assuming ZFC has a model).

For example, take a model 𝔐 of ZFC. Let 𝔐 satisfy the statement "S | S is uncountable." So, there is no bijection B ∈ 𝔐 from S to ℕ. Now, let's just add B to 𝔐 and let 𝔑 be 𝔐 ∪ B. Thus S ∈ 𝔑 is countable.

When there's a mathematical result such as Skolem's Paradox indicating the relativity of countability (at least with first-order formulations of ZFC, this problem does not come up in second-order formulations), we can ask, "Well, is countability actually relative?" Or some derivative questions: "What are the natural numbers, or what does it mean to say there is a bijection between the natural numbers in the real world?" (whatever you take to be "actual" or the "real world")?

More generally, what is the relation of math-objects/results to the real world (however construed)?

Also see the same question asked and answered on the Mathematics site.

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    possible duplicate of Was mathematics invented or discovered? – Michael Dorfman May 24 '12 at 6:54
  • Thanks for the links. I'll read them first before further posting. I'm just torn: on the one hand, I can write that "Cantor's Theorem tells us that uncountable sets exist" and be fine with it, but in perhaps what I take to me a more strict sense of existence, it makes no sense to me. – pichael May 24 '12 at 7:51
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    Your qualm is a reasonable one; the notion of existence is fraught with difficulty. What do we mean when we say that the following things exist? {the cup of water on my desk, a hurricane, Beethoven's Ninth Symphony, thunder, the color blue, justice, Huck Finn, the Pythagorean theorem, a quark, the letter 'e', the particular letter 'e' on my screen} – Michael Dorfman May 24 '12 at 8:54
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    Please keep in mind that copying and pasting your question onto multiple sites is strongly discouraged. If you think (as perhaps in this case) that the answers from two different perspectives would be useful, please take care to craft each question for the site you're posting it on. The version posted to Philosophy.SE should have a decidedly philosophical tone, whereas the version on Mathematics.SE should be more rigorously mathematical in nature. (Related: The global SE policy on cross-posting) – Cody Gray May 25 '12 at 8:00
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    No, if it's received helpful answers, there's nothing wrong with leaving it here. You can always edit the question with improvements. Just leaving a comment for future reference, as this got a couple of moderator flags for being a duplicate. – Cody Gray May 26 '12 at 8:09
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There were at least three different idealistic positions regarding the existence of mathematical objects:

  • Mathematical concepts like geometry or logic exist before any (physical) experience in our mind and are what enables us to have and interpret experiences. Mathematical objects exist a priory as categories of our mind.
  • Mathematical objects exist in a stronger and more real sense than physical objects. Physical objects only exist as shadows of ideal objects, while mathematical objects exist as ideal objects.
  • The existence or non-existence of mathematical objects tells us something about the existence or non-existence of actual or possible physical objects with corresponding properties.

Common to all three position is a strong commitment to the existence of mathematical objects and to the falsifiable consequences of that existence. These strong forms of idealism have actually been falsified theoretically and practically by progress on the foundations of mathematics and the tremendous success of abstract mathematics. (But the strong forms of formalism have also been falsified.)

You might want to consider some potential commitments related to strong forms of mathematical idealism: Foundations of mathematics are neither necessary nor possible. Axioms have to be intuitively true, like Euclid's axioms for geometry. Mathematical object are at least as real as any object in the physical world. The question of whether objects like zero, infinity, square root of two, or square root of minus one actually exist is non-trivial and must be answered for each of these objects separately.

Note The above description is neither historically correct nor does it give full justice to the corresponding positions. However, I preferred to give a description of idealistic positions which still includes its rough edges, as a contrast to the "why should we care" position of fictionalism.

  • What does it mean to say a mathematical object qua an abstraction exists? What kind of existence is it? – pichael May 27 '12 at 3:34
  • @pichael I rewrote the answer to be much more concrete about what mathematical idealism actually can mean. It turns out that this is quite different from current mathematical practice regarding abstract objects and abstraction in general. – Thomas Klimpel May 30 '12 at 8:27
  • Thanks for the rewrite! What are (or can you point to a resource) the consequences and the arguments for their falsification? The first seems wrong: much of math is counter-intuitive ("N and Z are the same size?! What?!" I like the third: I don't seem an immediate reason that math-results/objects could possibly be (materially) instantiated in nature. Although I don't know what material instantiation a higher transfinite would look like. – pichael May 31 '12 at 0:58
  • @pichael The first one tried to summarize Kant's idealism. It's not important whether some mathematical results are counter-intuitive, the important point is that they exist a priory before any (physical) experience. An argument for its falsification might be the Church-Turing thesis, because an universal turing machine doesn't need many preexisting concepts. Strong platonists like Roger Penrose therefore reject this thesis. – Thomas Klimpel May 31 '12 at 22:18
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There are a number of schools of thought on this:

Mathematical Platonists argue that mathematical objects exist in the same way that other Platonic ideals exist.

Intuitionists, on the other hand, argue that mathematical objects exist only as psychological constructions.

Fictionalists argue that it doesn't really matter if mathematical objects exist or not-- that one can bypass ontological commitments altogether by treating mathematical objects as convenient fictions.

Personally, I'd suggest you start with the question raised by the Fictionalists, and ask yourself: why does it matter?

  • I read the "schools of thought" link (SEP: Philosophy of Math) and the nominalist structuralists position most resonated with me. However, it doesn't seem immediately obvious that it is possible for higher transfinite cardinalities to be instantiated in nature. – pichael May 27 '12 at 3:30
  • @pichael: if by "instantiated in nature", you mean "taking a material form", I agree completely-- the universe appears by all observation to contain a finite number of particles. If by "instantiated in nature" you mean "conceivable within the imagination so that we can perform abstract mathematical operations upon them", I don't see why not. – Michael Dorfman May 27 '12 at 9:44
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    Hmmm...I had initially meant just "taking a material form," but I see that I must take a less restricted view of "instantiated in nature." I'm (in) nature, and I conceive and imagine things. Where else could this be "going-on" other than in nature? – pichael May 31 '12 at 0:04
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Numbers on their own represent nothing, one always represents an entity, two and beyond are all groups of entities. It is a matter what entity the number represents. If that entity is for example millimeters then it is an objective unit of measurement, if that unit is some arbitrary mental unit it relates to nothing and the sum is nonsense.

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    Do complex numbers also represent entities? What entity does 1/2 represent? – pichael May 27 '12 at 3:37

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