The Nature of Abstract Mathematical Objects (II)
We cannot suppose that changing the name of an otherwise concrete entity and downgrading its status to the realm of the abstract will provide us with any better understanding of the epistemology of platonic objects, but I find equally unreasonable to assume that the concrete and the abstract categories are ontologically fixed attributes that every object is attached to independently of the over-all structure of our theories.
How does all this bear on the problem of the epistemological access to abstract entities, and in particular to mathematical platonism? I would argue that not in a direct way, but it can serve to exemplify a case where some knowledge of abstract entities is possible and therefore that can be translated (this is my main thesis) to the case of mathematical objects. In the first place we should enquire whether there can be different grades of “abstraction”. Are there partially abstract objects or else is there a sharp and well defined distinction between the abstract and the concrete, as Burgess and Rosen defend [9, p.14]? As it has been shown in the case of microscopic systems not “occupying space”, I sustain that such distinction is not fundamental and what can be considered quasi-abstract in one scenario may turn out to be concrete in a different configuration. Using an alterative definition in terms of what it means for an object to occupy space in relation to another object, we can have a better chance to explain the difficulties in drawing a distinctive line between the abstract and the concrete. Another area where this approach can be proved fruitful is in analyzing cosmological theories, like Max Tegmark’s ontic structural realism [10]. If the mathematical universe hypothesis [MUH] is correct, then all elements in the physical universe are isomorphic to elements in a mathematical structure, therefore it would be interesting to see if there is any point in the over-all scheme where physical relations become nothing more, and nothing less, than mathematical relations, i.e. where any remaining trace of substance gradually dissolves into an abstract form. I think we have suggested an argument in that direction.
Can any of these justify our belief in mathematical objects, or at least support the reliability of mathematical claims? Well, if the general argument against platonic objects is that, conceding that such objects exist, we can not have any empirical knowledge of them, it seems we have shown that we do have some empirical knowledge of quasi-abstract entities, and what is more important, of entities that share the same lack of spatio-temporal definiteness than mathematical entities (that is the reason why Field’s abstract points went so easily disguised as physical points after all). There may still be a final stretch that we need to trade in order to give a complete justification of our belief in those entities, but that seems not far beyond our reach once the indispensability argument has been supported by undermining Field’s nominalistic programme [11, p.245]. Maybe is not entirely incorrect to say that the “empirical” knowledge we have of mathematical objects comes just by its irreplaceable use in scientific theories all across the board.
If we have to choose between the primitive abstractness and the non-spatiality (or causal inefficacy) of mathematical objects in order to explain the particular nature of those objects [8], I argued that we should be inclined to defend their non-spatiality (and causal inefficacy). But the question that follows immediately is then, are there any other objects in nature (excluding the products of our imagination) which do not occupy space in a similar way and are causally contentious? The answer that has been defended here is yes, there are. And the empirical knowledge we can extract from them partakes of the fact that they are not absolutely abstract but only relatively so, their whereabouts supervene on what goes on in the macroscopic world. As long as their properties are indicated by macroscopic facts, those objects are as concrete as tables and chairs.
Mathematical objects bear a similar relation to the macroscopic world, which includes us human beings with all our forms of interaction and knowledge, as those non-spatial entities, though probably in a more radical way. And this is the key point, because how “radical” that departure is will more than likely be the main argument against this thesis. One way to meet that challenge would be to bite the bullet and accept the aforementioned radical departure, and then ask how two objects which do not occupy space in any sense (i.e. have no spatial extent) can differ in grade on that same attribute. Here is the only way in which I think they may differ, microscopic objects do not occupy space in relation to a world that is spatially richer than the microscopic realm, that world is called macroscopic, while mathematical objects do not occupy space in relation to any possible world. In a similar way, macroscopic objects do not occupy space only in relation to an imaginable world that is richer than then macroscopic realm, if that imaginable world is the one occupied by platonic objects that is an open question for the metaphysician.
Adrian Icazuriaga
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We cannot suppose that changing the name of an otherwise concrete entity and downgrading its status to the realm of the abstract will provide us with any better understanding of the epistemology of platonic objects, but I find equally unreasonable to assume that the concrete and the abstract categories are ontologically fixed attributes that every object is attached to independently of the over-all structure of our theories.
How does all this bear on the problem of the epistemological access to abstract entities, and in particular to mathematical platonism? I would argue that not in a direct way, but it can serve to exemplify a case where some knowledge of abstract entities is possible and therefore that can be translated (this is my main thesis) to the case of mathematical objects. In the first place we should enquire whether there can be different grades of “abstraction”. Are there partially abstract objects or else is there a sharp and well defined distinction between the abstract and the concrete, as Burgess and Rosen defend [9, p.14]? As it has been shown in the case of microscopic systems not “occupying space”, I sustain that such distinction is not fundamental and what can be considered quasi-abstract in one scenario may turn out to be concrete in a different configuration. Using an alterative definition in terms of what it means for an object to occupy space in relation to another object, we can have a better chance to explain the difficulties in drawing a distinctive line between the abstract and the concrete. Another area where this approach can be proved fruitful is in analyzing cosmological theories, like Max Tegmark’s ontic structural realism [10]. If the mathematical universe hypothesis [MUH] is correct, then all elements in the physical universe are isomorphic to elements in a mathematical structure, therefore it would be interesting to see if there is any point in the over-all scheme where physical relations become nothing more, and nothing less, than mathematical relations, i.e. where any remaining trace of substance gradually dissolves into an abstract form. I think we have suggested an argument in that direction.
Can any of these justify our belief in mathematical objects, or at least support the reliability of mathematical claims? Well, if the general argument against platonic objects is that, conceding that such objects exist, we can not have any empirical knowledge of them, it seems we have shown that we do have some empirical knowledge of quasi-abstract entities, and what is more important, of entities that share the same lack of spatio-temporal definiteness than mathematical entities (that is the reason why Field’s abstract points went so easily disguised as physical points after all). There may still be a final stretch that we need to trade in order to give a complete justification of our belief in those entities, but that seems not far beyond our reach once the indispensability argument has been supported by undermining Field’s nominalistic programme [11, p.245]. Maybe is not entirely incorrect to say that the “empirical” knowledge we have of mathematical objects comes just by its irreplaceable use in scientific theories all across the board.
If we have to choose between the primitive abstractness and the non-spatiality (or causal inefficacy) of mathematical objects in order to explain the particular nature of those objects [8], I argued that we should be inclined to defend their non-spatiality (and causal inefficacy). But the question that follows immediately is then, are there any other objects in nature (excluding the products of our imagination) which do not occupy space in a similar way and are causally contentious? The answer that has been defended here is yes, there are. And the empirical knowledge we can extract from them partakes of the fact that they are not absolutely abstract but only relatively so, their whereabouts supervene on what goes on in the macroscopic world. As long as their properties are indicated by macroscopic facts, those objects are as concrete as tables and chairs.
Mathematical objects bear a similar relation to the macroscopic world, which includes us human beings with all our forms of interaction and knowledge, as those non-spatial entities, though probably in a more radical way. And this is the key point, because how “radical” that departure is will more than likely be the main argument against this thesis. One way to meet that challenge would be to bite the bullet and accept the aforementioned radical departure, and then ask how two objects which do not occupy space in any sense (i.e. have no spatial extent) can differ in grade on that same attribute. Here is the only way in which I think they may differ, microscopic objects do not occupy space in relation to a world that is spatially richer than the microscopic realm, that world is called macroscopic, while mathematical objects do not occupy space in relation to any possible world. In a similar way, macroscopic objects do not occupy space only in relation to an imaginable world that is richer than then macroscopic realm, if that imaginable world is the one occupied by platonic objects that is an open question for the metaphysician.
Adrian Icazuriaga
Volver
