The Nature of Abstract Mathematical Objects (I)
Though epistemic knowledge of abstract mathematical objects is not free of difficulties, I shall argue that is not more problematic than the understanding of the nature of physical reality, at least if we include the realm of elementary “constituents” as being part of that physical reality. Not only particles but the so called “structure of space and time” shares the dubious existence of abstract mathematical objects.
The reason to focus on space-time points and regions in the search for an epistemic theory of mathematical objects is quite obvious. The non-spatiality (or non-spatiotemporallity) of abstract objects, together with their being causally ineficatious, is the recurring standard definition and the most prominent characteristic of those entities [8]. We may gain some understanding of their nature if we manage to see how other elements that share some of the properties of mathematical objects, in this case their non-spatiality, relate to the physical world of us human beings.
Let’s analyze the mentioned standard definition of abstract objects: “An object is abstract if it fails to occupy anything like a determinate region of space (or spacetime)” [8], but on a slightly modified version. An object that fails to occupy a determinate region of space is abstract, but only with respect to a region of space that is spatially richer than the region of space the object purportedly occupies. According to this alternative definition, which is indebted to the PIQM, macroscopic objects are not abstract in the actual world, only counterfactually against a possible world that can attribute sharper positions than the ones we experience. In a similar way, electrons and subatomic particles are quasi-abstract (and causally ineficatious), as long as their positions are not indicated by macroscopic facts, this is a somehow uncomfortable exception that has to be made. They are abstract with respect to the macroscopic world and they are causally ineficatiouns because they do not cause counters to click (there is no need to recur to causal notions if all we have is a probability algorithm that correlates elementary facts). This interpretation, though controversial, does not contradict the results of quantum correlation laws, and is not part of any interpretation of “esoteric physics” [9, p.20] as some philosophers may be inclined to think. This is simply what the laws of physics are telling us. There seems to be a conspicuous tendency among some philosophers to move inside standardized compartments which are specifically build to create disagreement and debate, as in fear that without that preclusion there may be little or nothing to talk about. “In checking against the list of paradigms and foils [of abstraction], arguably no great weight should be put on the wilder and more exotic examples from physics or methaphysics, such as quantum wavicles or possible worlds” [9, p.20]. I do not see any epistemological or methodological justification for such a claim, and one is left with the impression that some reasonable answers to our long-standing paradoxes may be found exactly in that “esoteric” realm.
I do not think that Rosen’s argument of impure sets as potentially existing in space and time [8] provides a counterexample to the previous definition of abstract objects in terms of trivial, though relative, spatiotemporal regions. Saying that “a set of books is located on a certain shelf in the library” is a way of talking about the books, not a way of talking about where the abstract set is. For example, the sentence “The Complete Works of Henry James are located on shelf H11” does not say that an abstract entity, namely The Complete Works of Henry James, exists at a definite spatial region, what it says is that a particular instance of Henry James’s works (the edition bought by this library) has a well defined spatial property. Similarly, saying that the set of my left shoes is located in the same place as the set of my right shoes may be just a way of referring to my wardrobe. And the same would apply to pure sets, like {0, {0}}, illustrated by the sentence: “the set corresponding to the third element of von Neumann’s ordinals is located on this page”, that set just represents an instance of von Neumann’s ordinals containing two elements, which as a matter of fact are on this page. When we say that a set containing two concrete elements exists at a certain region of space we are saying something about those two individual elements, a shortening for saying that this element and that element and there, but when we say that a set containing two concrete elements exists wherever those concrete elements are located we are making an invalid existential predication, is not a way of talking about the concrete and is not a way of talking about the abstract either.
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Though epistemic knowledge of abstract mathematical objects is not free of difficulties, I shall argue that is not more problematic than the understanding of the nature of physical reality, at least if we include the realm of elementary “constituents” as being part of that physical reality. Not only particles but the so called “structure of space and time” shares the dubious existence of abstract mathematical objects.
The reason to focus on space-time points and regions in the search for an epistemic theory of mathematical objects is quite obvious. The non-spatiality (or non-spatiotemporallity) of abstract objects, together with their being causally ineficatious, is the recurring standard definition and the most prominent characteristic of those entities [8]. We may gain some understanding of their nature if we manage to see how other elements that share some of the properties of mathematical objects, in this case their non-spatiality, relate to the physical world of us human beings.
Let’s analyze the mentioned standard definition of abstract objects: “An object is abstract if it fails to occupy anything like a determinate region of space (or spacetime)” [8], but on a slightly modified version. An object that fails to occupy a determinate region of space is abstract, but only with respect to a region of space that is spatially richer than the region of space the object purportedly occupies. According to this alternative definition, which is indebted to the PIQM, macroscopic objects are not abstract in the actual world, only counterfactually against a possible world that can attribute sharper positions than the ones we experience. In a similar way, electrons and subatomic particles are quasi-abstract (and causally ineficatious), as long as their positions are not indicated by macroscopic facts, this is a somehow uncomfortable exception that has to be made. They are abstract with respect to the macroscopic world and they are causally ineficatiouns because they do not cause counters to click (there is no need to recur to causal notions if all we have is a probability algorithm that correlates elementary facts). This interpretation, though controversial, does not contradict the results of quantum correlation laws, and is not part of any interpretation of “esoteric physics” [9, p.20] as some philosophers may be inclined to think. This is simply what the laws of physics are telling us. There seems to be a conspicuous tendency among some philosophers to move inside standardized compartments which are specifically build to create disagreement and debate, as in fear that without that preclusion there may be little or nothing to talk about. “In checking against the list of paradigms and foils [of abstraction], arguably no great weight should be put on the wilder and more exotic examples from physics or methaphysics, such as quantum wavicles or possible worlds” [9, p.20]. I do not see any epistemological or methodological justification for such a claim, and one is left with the impression that some reasonable answers to our long-standing paradoxes may be found exactly in that “esoteric” realm.
I do not think that Rosen’s argument of impure sets as potentially existing in space and time [8] provides a counterexample to the previous definition of abstract objects in terms of trivial, though relative, spatiotemporal regions. Saying that “a set of books is located on a certain shelf in the library” is a way of talking about the books, not a way of talking about where the abstract set is. For example, the sentence “The Complete Works of Henry James are located on shelf H11” does not say that an abstract entity, namely The Complete Works of Henry James, exists at a definite spatial region, what it says is that a particular instance of Henry James’s works (the edition bought by this library) has a well defined spatial property. Similarly, saying that the set of my left shoes is located in the same place as the set of my right shoes may be just a way of referring to my wardrobe. And the same would apply to pure sets, like {0, {0}}, illustrated by the sentence: “the set corresponding to the third element of von Neumann’s ordinals is located on this page”, that set just represents an instance of von Neumann’s ordinals containing two elements, which as a matter of fact are on this page. When we say that a set containing two concrete elements exists at a certain region of space we are saying something about those two individual elements, a shortening for saying that this element and that element and there, but when we say that a set containing two concrete elements exists wherever those concrete elements are located we are making an invalid existential predication, is not a way of talking about the concrete and is not a way of talking about the abstract either.
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