Abstract
Structures: where a particular transcendental-phenomenological brand of structuralism is presented. I discuss, in particular, how structures can be investigated by being interpreted in other structures, which plays a central role in my approach to the problem of the applicability of mathematics. I also confront my approach with more traditional structuralist perspectives in the philosophy of mathematics.
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- 1.
“La Mathématique estl’ art de donner le même nom à de choses différentes. Il faut s’entendre. Il convient que ces choses, différentes par la matière, soient semblables par la forme, qu’elles puissent, pour ainsi dire, se couler dans le même moule [...]. Dans un group la matière nous intéresse peu, [...] c’est la forme seule qui importe, [...] quand on connaît bien un group, on connaît par cela même tous les groups isomorphes.” (Poincaré 1908).
- 2.
Ontological parsimony is not in itself desirable if it is misplaced or plainly wrong.
- 3.
One can define functions as particular relations: if f is a n-ary function defined in a domain D, one can define a (n + 1)-ary relation R such that R(x 1 , …, x n , y) iff y = f(x 1 , …, x n ). I say that a subdomain B of D is closed under f iff for every sequence x 1… x n of elements of B, y = f(x 1 , …, x n ) is also an element of B. In general, when considering subdomains where functions are defined one supposes they are closed under these functions. To restrict a n-ary relation S defined in a domain D to a subdomain B of D is to consider the n-relation S′ defined in B thus: for all sequences x 1 , …, x n of elements of B, S′(x 1 , …, x n ) iff S(x 1 , …, x n ). If R is defined as above, its restriction R’ in B is such that for all sequences x 1 … x n , y of elements of B, R’(x 1 ,… x n , y) iff y = f(x 1 , …, x n ). There may exist elements x 1 … x n of B but no y in B such that R’(x 1 ,… x n , y) unless B is closed under f.
- 4.
Again, there is a phenomenological difference between the structure as abstract (i.e. non-independent) aspect of a particular domain, which is only an aspect of this domain, and the structure as an ideal entity, which is not an aspect of anything, but an object indifferently instantiable as the abstract aspect of materially distinct, but formally equal, domains. The ideal structure is the type, all the equal abstract structures the tokens.
- 5.
If a categorical structure positing theory is incomplete, and some sentence undecidable in the theory is proven in a consistent extension of it, then the categoricity of the original theory implies that this sentence is true in the structure it posits. This is a trivial example of how one can investigate the structure posited by a theory by means other than the positing theory itself.
- 6.
See, for instance, Descartes’ Geometry for how to operate with numbers geometrically.
- 7.
See Shoenfield 1967, chap. 5, for details.
- 8.
Hilbert 1900.
- 9.
The question of whether operating with imaginary entities produces always true results about real entities depend on how both operational domain, the narrower (real) and the larger (imaginary), relate to one another. See below.
- 10.
Pierre-Louis Wantzel, in 1837
- 11.
Niels Abel, in 1824, showed that there are equations of the fifth degree that were not solvable by radicals. However, it was Évariste Galois (died in 1932 aged 20) who developed the general theory known today as Galois Theory, where the quintic is shown to be in general unsolvable by radicals. Galois was the first to use the word “group”; today we consider him one of the founders of abstract algebra.
- 12.
For a detailed analysis of the rise of mathematical structures, see Corry 1996.
- 13.
“Being (Seiend) in the broadest sense, in that of theory of science and formal ontology, is each and every thing that can figure as the subject of a statement, each and every thing about which we can in truth speak, each and every thing that can in truth be referred to as being (seiend)” (Husserl 2008).
- 14.
For this reason, Husserl placed formal mathematics, insofar as their objective correlates are concerned, in the realm of formal ontology, the ontological side of the third realm of the three-stored edifice of formal logic (the other side, the apophantic, concerns itself with theories and logical relations among them).
- 15.
Again, since the elements of N’ are sets, it is true (in set theory) that ∅ ∈ {∅}, but this does not express a structural property of N’; it has no sense as a property of the ω-sequence.
- 16.
In mathematics, formal theories are typically introduced thus: “let there be a domain of objects, no matter what they are, where certain unspecified relations are defined, no matter how, such that …”, the blank being filled with the formal axioms of the theory. If this theory is consistent and categorical, it posits a structure (i.e. it brings it to intentional existence) whose structuring relations, albeit materially indeterminate, must obey the formal stipulations established in the axioms.
- 17.
Keeping in mind that this presupposing is part of the intentional constituting act, not a hypothesis.
- 18.
“Abstract” simply means “non-independent”. Husserl’s explanation of the concepts of dependent existence and abstract objects can be found in the Logical Investigations, 2nd Investigation for abstraction, and 3th Investigation for the notion of ontological dependence.
- 19.
When a mathematician says, for instance, “let us consider (or imagine) the system of all motions in space with composition as the structuring operation”, he is positing a structured system of entities. What in this way is intended is not each of the infinitely many motions individually, but the operational domain as a whole, with the intuitive properties associated with its ruling concept (for instance, motions are continuous rigid point transformations that can be reversed to cancel themselves). The intuition of a mathematical domain (and imagining is intuiting) does not require the intuition of all of its elements individually, but, in most cases, the intuition of a concept (in our case, rigid motions in space) under which these materially determined elements fall. To think of mathematical intuition as the intuition of objects and to construe the intuition of a mathematical manifold as the summation of the intuition of its elements individually betrays the wrong conception that mathematics is a science of objects. The adequate intuition of the objects of a mathematical domain individually is not a necessary condition for the intuition of the structure underlying this domain. We can furthermore abstract the structure of the system of motions in space (a particular group) by ignoring the nature of its entities or the nature of the operation structuring it and concentrating only on its formal properties. By considering generically domains of unspecified entities structured by unspecified binary operations having the properties of associativity, existence of compositionally neutral elements and inverses, we move to a higher level of theoretical interest, a general theory of abstract structures of a type (group theory in this case).
- 20.
Of course, I do not claim that any consistent theory has a set-theoretical model in the sense of model theory; I am not particularly interested in this type of models. From my phenomenological perspective, any consistent formal theory posits a formal domain (although there are – second-order – consistent theories that do not have set-theoretical models), whose existence the consistency of the theory is sufficient to grant, and that is accessible only through the theory. The fact that domains such as these do not have material instantiations does not mean that they do not exist.
- 21.
See Shoenfield 1967, pp. 61–65. This definition could be generalized by dropping the requirement that T’ is categorical, i.e. that it is itself a structure-positing theory.
- 22.
This strategy is widely used in mathematics and mathematical physics. In this last case, it sometimes pays off to take logical consequences in purely mathematical extensions of mathematical models of empirical reality as maybe expressing facts about reality (even though mathematical theories of reality may not be categorical). Here, logic gives place to heuristics.
- 23.
In fact, the new formal object is originally only a symbol; it acquires “formal objecthood” only when the way it relates operationally with the other formal objects of the domain where it is introduced is determined. With the introduction of this new object the entire domain, and the objects in it, change, they become other objects and another domain.
- 24.
“The true value of such numbers [complex numbers] lies in the fact that they enable us to form connections between entirely different parts of mathematics.” (Waismann 2003).
- 25.
da Silva 2016a.
- 26.
Resnik 1981, p. 530.
- 27.
Commenting on Resnik’s characterization of structuralism, C. Parsons (1990) says that structuralism “is most persuasive [...] in the case of pure mathematical objects such as sets and numbers […]. In these cases, we look in vain for anything else to identify them beyond basic relations of the structure to which they belong […]”. What about the concept of number as quantitative form? Numbers proper are something; they fall under a well-determined (and intuitable) concept; telling how numbers relate to one another is not telling what numbers are.
References
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Da Silva, J. J. (2016b). The analytic/synthetic dichotomy. Husserl and the analytic tradition. In G. E. Rosado Haddock (Ed.), Husserl and analytic philosophy (pp. 35–54). Berlin: De Gruyter.
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da Silva, J.J. (2017). Structures. In: Mathematics and Its Applications. Synthese Library, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-319-63073-1_7
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