Abstract
In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Book M, 1076a.
Husserl’s account of the constitution of numbers can be found in the Philosophy of Arithmetic (from a psychological, not yet transcendental perspective), and the corresponding (transcendental) theory for geometrical entities in “The Origin of Geometry”, an appendix to The Crisis of European Sciences and Transcendental Phenomenology (1970). Husserl’s general theory of intuition can be found in his Logical Investigations (2001) and Ideas I (1931). There is an extensive secondary literature on this topic, including Rosado Haddock (1987), Miller (1982), Tragesser (1984), Levinas (1984), among others.
With some good will we can see Maddy’s theory of set perception (Maddy 1980) as a naturalistic version of this phenomenological truism; but, in any case, it is difficult to see how the perception of singletons, for instance, can be due to a “set detector” in the brain, whereas, on the other hand, the phenomenological difference between perception of elements and perception of their singletons is evident. Also, we can make any mereological sum into a set by an intentional act, but it is hard to see how brain set detectors can be so condescending. In short, naturalizing phenomenology is not an easy task.
By pure numbers I mean numbers that do not depend on the nature of the elements of the collections they number (but only on their quantity).
Husserl, in his Philosophy of Arithmetic, criticized Frege for basing the definition of number on the relation of equinumerosity; he thought (with Cantor, who probably gave him this idea) that we could intuit numbers directly by abstracting them out of given collections, and then define equinumerosity in terms of numerical identity. He later withdrew this criticism.
We can ignore this order because it is irrelevant for the cardinality of collections, although we usually establish cardinality by counting, i.e., by establishing first ordinality (to count is to establish a 1-1 correspondence between the collection being counted and an ordinal). This causes no discrepancy in the finite case, for the counting of finite collections, no matter how it is done, always yields the same ordinal number, which coincides with its cardinality. The phenomenological difference in the constitution of ordinal and cardinal numbers should not concern us here.
See, for instance, Barrow 1993, chap. 2.
Plato’s theory of numbers (if we accept Aristotle’s account of it) distinguished between ideal and mathematical numbers; the former were, for him, Ideas; the latter, collections of monads, which is nothing but abstract quantitative forms.
To idealize is to actualize a possibility, to actually reach a limit point. It is a fundamental constituting act in mathematics, by mean of which we obtain, for instance, infinite collections, perfect figures, real numbers (as limit points of Cauchy sequences or Dedekind cuts), the whole system of Cantor’s infinite ordinals and cardinals or choice sets. We can characterize constructivism in mathematics as the refusal, to some extent, to idealize.
Some definitions: a structured domain (a system, according to Shapiro 1997) is a collection (not necessarily a set) of objects and structuring relations (where by “object” I mean any bearer of attributes, or, from a linguistic perspective, the referent of any nominal term of a language). Structural properties are properties of a domain that remain true in arbitrary isomorphic copies of it. Let me make this clearer. Once its structuring relations or, equivalently, its structure is fixed, some of the properties of a domain may not be structural (even though any property of a domain is a structural property of some structuring of its universe). For example, let 0, 1, 2, etc., denote the initial elements of any arbitrary ω-sequence (discrete linear sequences obeying second-order Peano axioms). Consider the two isomorphic ω-sequences given, respectively, by Zermelo and Von Neumann finite ordinals: ∅, {∅}, {∅, {∅}}, …, and ∅, {∅}, {{∅}}, …; then 0 ∉ 2 in the second case, but 0 ∈ 2 in the first. So, although true in some ω-sequences, the assertion 0 ∈ 2 does not express a structural property of ω-sequences. Structural properties of a structured domain are then structural properties of all domains isomorphic to it; so, if we believe that structural properties are properties of structures, it is natural to define a structure as the common aspect of all mutually isomorphic domains (and structural properties of any domain instantiating the structure as properties of the structure). We must be careful here. It is a common mathematical practice to think of this “common aspect” as an isomorphism type, usually construed as a class of isomorphic domains. The existence of structures turns out then to depend on the existence of classes. I prefer to go the old-fashioned way, and think of structures as abstracta, i.e., correlates of experiences of abstraction, understood, however, not as psychological processes, but intentional experiences. What I mean is that structural descriptions (expressing structural properties) can be considered as either descriptions of particular structured systems, or, with a change of intentional drive, descriptions of structures. This “change of intentional drive” is just another name for abstraction, which becomes manifest in the fact that structural properties can be expressed formally, i.e., in non-interpreted languages. To abstract, as I understand the term here, means simply to focus on formal-structural properties of structured domains. So, structures are ontologically dependent on structured domains and acts of abstraction (which, as seen, has nothing to do with Frege’s universal solvent) and identification. There is another way of bringing structures to consciousness and referring to them, another type that is of intentional experiences in which structures are intended, namely, theorizing. But, as we will see below, it is not always the case that formal mathematical theories can characterize structures uniquely (in most cases, they can only characterize families of structures). It is not the case either that a theory can always completely describe a structure, even when it can characterizes it uniquely—more about this below.
If the structure of the domain of lions (including its structuring relations) had been completely determined, and we found a domain of flowers isomorphic to it, then we could learn about lions by studying flowers. The problem, of course, is that zoology has not so far accomplished such a determination, and even if it had, there is no guarantee that the domain of lions would have isomorphic copies more amenable to scientific investigation. So, if we want to know about lions, we must study lions; in this consists the difference between formal and material sciences. In mathematics we can solve geometrical problems by solving algebraic equations; in physics analogous techniques are available; in material sciences in general, however, they are much less disseminated.
Formal theories, when freely invented, such as quaternion arithmetic, more than descriptions, are stipulations—a sort of fiat. Of course, formal arithmetic is not a free creation, but, as already explained, the form of contentual arithmetic. But this is mathematically irrelevant; as far as mathematics is concerned it would not make any difference if it were.
Husserl says that a formal domain is the objective correlate of a formal theory.
An interpretation is only a possible material determination of a formal manifold: to determine a formal domain materially is to endow it with material features its theory does not predetermine nor exclude.
I do not attribute this view to Husserl. In fact, Husserl believed there is a difference between contentual and purely formal arithmetic. The former, he believed, is a science; the latter is not, and only becomes one when materially interpreted.
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 indeterminate, must obey the formal stipulations established in the axioms.
“Abstract” can simply mean “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 3rd Investigation for the notion of ontological dependence.
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 so as 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 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).
It is part of the noematic Sinn of empirical theories that their objects are real; so, no non-real object can be framed by this Sinn. The intentional objects of empirical theories are real objects and intentional existence is, in this case, real existence; if there are no real objects satisfying the noematic Sinn of an empirical theory, this Sinn is inconsistent and the positing experience simply cancels itself.
I do not, of course, claim that any consistent theory has a set-theoretical model in the sense of model theory; we are not particularly interested in this type of models. From our phenomenological perspective, any consistent formal theory posits a formal domain, whose existence the consistency of the theory is sufficient to grant, and which can be accessed only through the theory. This domain does not have, obviously, an independent existence, but this does not mean that it does not exist at all.
The structure of a formal domain is not necessarily completely determined if the theory is not categorical (even if it is complete). The problem is that non-categorical theories can have non-isomorphic realizations displaying structural differences that are not, however, “captured” by these theories.
By this I mean that the structuring relations of the system must be clearly stipulated.
I assume we can cognitively access structured systems once they are given to us, but I will not insist too much on how such systems can be “given”. By “directly given” I mean given by its elements and their structuring relations. Finite structured systems seem to pose no problem to be so given, provided their elements can be exhaustively displayed. Large finite and infinite structured systems, on the other hand, although able only to be partially displayed under our eyes, are usually endowed with a generation-and-limitation principle—somehow associated with the concept circumscribing these domains—on the basis of which they can be given. Given, for example, a finite linear discrete sequence of “points” followed by three dots, like this: xxxx …, it can only be “seen” as a ω- (as opposed, for instance, to a 2ω-) structured set on the basis of a generation principle: any point is followed by a next, and a limitation principle: going to the next is the only generation principle (for instance, we cannot take “limits”). Directly “perceiving” large or infinitary structured systems on the basis of some partial finite segment depends on preconceptions regarding what we perceive. In most cases, however, we are not given structured systems directly, but only indirectly by their ruling concepts, which are what we cognitively access.
Categoricity and completeness are, in general, relatively independent concepts; there are categorical incomplete theories, such as second-order arithmetic, and complete non-categorical theories, such as any complete first-order theory of an infinite mathematical domain—i.e., the set of all assertions of some first-order language that are true in an infinite domain. However, categoricity (and even categoricity in a power, for powers at least as big as the cardinality of the language of the theory) implies completeness for first-order theories. Categoricity always implies semantic completeness (a theory is semantically complete when all of its models satisfy the same sentences of the language of the theory; i.e., no two models of the theory can be distinguished within the expressive power of its language), but may not imply syntactic completeness.
Provided the domains of concern are enriched with extra structure, so as to fall under the theories we apply, and the original structures are related to their structural enrichment in some convenient way.
Formal and material are antithetic notions; the former denotes what does not depend on the particular nature of the objects and structuring relations considered, the former, what does depend on it. The dichotomy formal-material can be construed in terms of others, such as public–private, linguistically expressible-ineffable or even understanding-intuition; see appendix.
According to Weyl (1963, p. 113), even intuitive space and time must “be relinquished as building material [of the external world, JJS]; [and] replaced by a four-dimensional continuum in the abstract arithmetical sense. […]. What remains is ultimately a symbolic reconstruction of exactly the same kind as that which Hilbert carries through in mathematics.” For Weyl, the objective world is capable of being represented only symbolically, for what is objective is what remains after the subjective material content of our impressions is eliminated; i.e., only purely formal or structural relations, which can be adequately represented symbolically, are objective. He continues: “a systematic scientific explanation […] will erect the world of symbols as a realm by itself and then […] attempts to describe the relations that hold between the symbols representing objective conditions, on the one hand, and the corresponding data of consciousness, on the other.”
Of course, physical causal explanations are never final; there are always further hidden causal links.
Choosing numbers to represent the relevant aspects of the phenomenon betrays the privilege accorded to its formal aspects; for, as Weyl tells us, numbers are simply symbols representing formal relations: “if any two time-points O and E are given such that O is earlier than E, it is possible to fix conceptually further time-points P by referring them to the unit distance OE. This is done by constructing logically a relation t between three points such that for every two points O and E, of which O is the earlier, there is one and only one point P which satisfies the relation t between O, E and P, i.e., symbolically OP = t·OE. Numbers are merely concise symbols for such relations, defined logically from the primary relations” (Weyl 1952, p. 8; see also Weyl 1994). When representing our intuitive experience of movement by a numerical correlation between numbers we are then privileging formal aspects of our experience. It is essentially because modern physics (in the tradition Husserl called Galilean) made this option that mathematics can play so important a role in it.
This amounts to transferring to space and time the topological and metrical properties of their numerical representatives. In so doing we establish that those are the topological and metrical properties appropriate for a scientific treatment of phenomena in space and time. Depending on whether the property transferred has or does not have empirical support, it is, respectively, either an a posteriori or an a priori property of space or time. For instance, continuity of space is arguably a priori; tri-dimensionality, a posteriori.
Notice that we can hardly expect to tell precisely how position in space relates to position in time independently of numbering positions in space and time and transferring the problem to finding a function between these two sets of numbers. Mathematics not only contributes to the solution of the problem; it also, and in an essential way, contributes to the formulation of the problem: in this case, it tells us what a correlation among positions in space and time is supposed to be.
There is a detailed critical analysis of the mathematization of science and what Husserl called Galilean methodology in Husserl (1970). Weyl reminds us that even physical geometry has a purely formal character: “geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by ‘states of addition-machines’ and ‘gas-mixtures’ and ‘systems of solutions of linear equations’ […] we must recognize with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial” (Weyl 1952, p. 26).
This, I believe, is part of the explanation of the heuristic value of formal similarities that Steiner (1998) found so puzzling.
Dictionary of Scientific Biographies (Bohr)—Brazilian edition, Contraponto: Rio de Janeiro, 2009.
This is mutatis mutandis also valid for formal-symbolic theories, which, as we have already discussed, can only be true by describing structural features of their models.
Incidentally, in similar way, Heisenberg justifies religious ‘there is’, i.e., God can, for him, be vindicated as a useful (formally adequate in my terminology) “imaginary” entity.
“Beyond the knowledge gained from the individual sciences, there remains the task of comprehending. In spite of the fact that the views of philosophy sway from one system to another, we cannot dispense with it unless we are to convert knowledge into a meaningless chaos” (Weyl 1952, p. 10).
Quine, whose argument for the inscrutability of reference seems to amount to the same conclusion, argues explicitly for the view that scientific knowledge is always only structural knowledge: “[…] if we transform the range of objects of our science in any one-to-one fashion, by reinterpreting our terms and predicates as applying to new objects instead of the old ones, the entire evidential support of our science will remains undisturbed” (Quine 1992: 8; see also notes 27 and 32 above). And, incidentally, aren’t all the troubles faced by cognitive scientists trying to give third-person descriptions of first-person qualia a sign of the fundamental impossibility of the task?
References
Barrow J (1993) Pi in the sky. Penguin Books, London
Boltzmann L (1902) “Model”. Enciclopaedia Britannica, 10th edn. Adam and Charles Black, London
Heisenberg W (2007) Physics and philosophy. Harper, New York
Hertz H (1956) The principles of mechanics presented in a new form. Dover, New York
Husserl E (1931) Ideas: a general introduction to pure phenomenology. Allen & Unwin, London
Husserl E (1970) The crisis of European sciences and transcendental phenomenology. Northwestern University Press, Evanston
Husserl E (2001) Logical investigations, vol 2. Routledge, London
Levinas E (1984 (1930)) Théorie de l’intuition dans la phénoménologie de Husserl. Vrin, Paris
Maddy P (1980) Perception and mathematical intuition. Philos Rev 89:163–196
Miller JP (1982) Numbers in presence and absence. M. Nijhoff, The Hague
Poincaré H (1908) L’avenir des Mathématiques. Rev Gen Sci Pures Appl 18:930–939
Quine WVO (1992) Structure and nature. J Philos LXXXIX(1):5–9
Rosado Haddock G (1987) Husserl’s epistemology of mathematics and the foundations of platonism in mathematics. Hesserl Stud 4(2):81–102
Schlick M (1915) Die philosophische Bedeutung des Relativitätsprinzips. Z Philos Philos Krit 159:129–175
Shapiro S (1997) Philosophy of mathematics—structure and ontology. Oxford University Press, Oxford
Steiner M (1998) The applicability of mathematics as a philosophical problem. Cambridge University Press, Cambridge
Tragesser R (1984) Husserl and realism in logic and mathematics. Cambridge University Press, Cambridge
Weyl H (1952) Space, time, matter. Dover, New York
Weyl H (1963) Philosophy of mathematics and natural science. Atheneum, New York
Weyl H (1994) The continuum. Dover, New York
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
1.1 Structuralism without structures
In the main text above I favored the view that structures depend ontologically either on structured systems or formal (categorical) theories. To be honest, this perspective has a few shortcomings. Firstly, there is a clear asymmetry on how structures are intended; more or less directly (depending on how directly structured systems are given), as abstract aspects of structured systems, or indirectly, as intentional objects framed by formal theories (expressing the intentional meaning—the noematic Sinn—of structure-positing experiences). Secondly, although, in the first case, structures are always instantiated, in the second they may not be. As you may recall, structures posited by formal theories are said to exist provided these theories are logically consistent. There are, however, formal theories (formulated, for instance, in second-order languages) that are consistent, but do not posses any models. So, the structures these theories posit cannot be instantiated in structured sets.
For Shapiro’s ante rem structuralism (structures exist “out there” independently) theories only posit structures if they are coherent, where by coherence he means a condition stronger than consistency, which guarantees the existence of structures satisfying the theory. My way out was to insist on logical consistency to the best of our knowledge (which is what I call coherence) as the condition of existence, and allow non-instantiated structures to exist, i.e., structures existing only as objective correlates of their theories (and living or dying with them).
However, in this case, the ties between theories and the structures they posit are so tight that we may wonder whether there is any real difference between them. Phenomenologically, we may construe this difference as that existing between the intentional object of an experience (the determinable x, in Husserl’s somewhat cryptic terminology) and the sense with which it is intended, i.e., the noematic sense (expressed linguistically). So, why not drop structures altogether and consider only formal theories, i.e., structural descriptions?
We could still differentiate between structural descriptions proper, i.e., formal theories obtained by formally abstracting contentual theories, and improper structural descriptions or structural stipulations, otherwise. We could even drop the requirement of categoricity; preserving, however, that of logical consistency (to the best of our knowledge). If not actually obeyed in some structured system, structural stipulations would be nothing but theoretical fictions. The somehow surprising fact is that they could, nonetheless, be of great scientific interest and utility.
Let T be a coherent structural description that may not happen to describe any actually or even possibly existing structure, and P the description of an actually existing structure S (or, equivalently, the structural description of an actual structured system S; for instance, a chunk of physical reality). Suppose that T extends P consistently (the language of T can be the same or an extension of the language of P, in which case T is said to involve “imaginaries”) and that a formula τ of the language of P is provable in T. Question: is τ true or false in S? Since T is a consistent extension of P, P cannot prove its negation, but we are still in the dark as to the truth of τ. It does not follow that τ is false nor does it follow that it is true in S, unless T is a conservative extension of P or P is syntactically complete (this is essentially the—correct!—solution given by Husserl to the problem posed by the use of “imaginaries” in contentual mathematics).
We can nonetheless conjecture that τ is true and seek to verify this conjecture by a closer scrutiny of S (or P). In case τ turns out to be false, we add ¬τ to P (so improving the theory P); if it is true, T played an important heuristic role in the scientific investigation of S. Interplay of theoretical speculation involving, to some extent, “fictional” theories and direct observations is a good description of the methodology of physics. In case τ turns out to be true, this, of course, would not imply the existence of any mysterious link between S and T (or we, the people who designed T)—after all, for most T’s τ would turn out to be false. Conclusion: scientific “fictions”, provided they are well designed, have a major role in science (against one line of argumentation of so-called “indispensability arguments”). But, of course, the design of good T’s, i.e., theories able to uncover true properties of S, require scientific acumen and deep understanding of mathematical relations among theories (this is why Husserl though the development of such an understanding to be the highest and most abstract aspect of formal logic).
Consider now the following, pertinent question: what a structured system of objects has over and above its structure? The obvious answer is, of course, its material content, i.e., particular objects and particular relations among them. But how can we single out descriptively (linguistically) this content? There is no other way but denoting objects by singular terms and relations by relational terms and writing true sentences involving these terms, in the hope that these sentences might single out (fix) the content in question. But no matter how meticulous this description is, it will always be true in other materially different, identically structured systems. So, the conclusion imposes itself that the particular material content of given structured systems cannot be captured descriptively, in language, but only intuitively, privately and ineffably. Our experience of the world, insofar as it is communicable and capable of being objectified is confined to its structural aspects; the particular material content of the world can only be object of private experiences, essentially incommunicable.Footnote 38 Descriptive sciences such as Goethe’s theory of colors, whose aim is to capture in words the essence of color experiences, are bound to be at best only approximate and immune to mathematical treatment (reality then dividing between privately accessible material content and publicly accessible structure). Modern science, which privileges the objective, that is, the structural aspects of the world, was a necessary outcome of our attempt to rigorously capture our experience of the world in words, and therefore, symbols; the mathematization of science was the crowning of the process. In conclusion, any description is a structural description.
From this perspective, mathematics is essentially formal (even contentual theories, whose mathematical content is entirely preserved under formal abstraction into equiform formal theories), applicable in science to the extent of its ability to provide theories adequately describing our experience of the (necessarily structural aspects of the) world, when conveniently interpreted, or displaying mathematically interesting and heuristically rich connections with theories chosen as convenient (structural) descriptions of the world.
Rights and permissions
About this article
Cite this article
Silva, J.J.d. Structuralism and the Applicability of Mathematics. Axiomathes 20, 229–253 (2010). https://doi.org/10.1007/s10516-010-9102-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10516-010-9102-3