structure for 'Mathematics'    |     alphabetical list of themes    |     unexpand these ideas

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism

[structuralism with real objects or real structures]

12 ideas
There are too many mathematical objects for them all to be mental or physical [Resnik]
     Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1)
     A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology.
Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
     Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1)
     A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject.
Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]
     Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3)
     A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component?
Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
     Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4)
     A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns.
Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
     Full Idea: Because the same structure can be exemplified by more than one system, a structure is a one-over-many.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
     A reaction: The phrase 'one-over-many' is a classic Greek hallmark of a universal. Cf. Idea 10217, where Shapiro talks of arriving at structures by abstraction, through focusing and ignoring. This sounds more like a creation than a platonic universal.
There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
     Full Idea: There is no 'structure of all structures', just as there is no set of all sets.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.4)
     A reaction: If one cannot abstract from all the structures to a higher level, why should Shapiro have abstracted from the systems/models to get the over-arching structures?
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
     Full Idea: Shapiro's structuralism champions model theory as the branch of mathematics that best describes mathematics. The essence of mathematical activity is seen as an exercise in comparing mathematical structures to each other.
     From: report of Stewart Shapiro (Philosophy of Mathematics [1997], 4.4) by Michèle Friend - Introducing the Philosophy of Mathematics
     A reaction: Note it 'best describes' it, rather than being foundational. Assessing whether propositional logic is complete is given as an example of model theory. That makes model theory a very high-level activity. Does it capture simple arithmetic?
To see a structure in something, we must already have the idea of the structure [Brown,JR]
     Full Idea: Epistemology is a big worry for structuralists. ..To conjecture that something has a particular structure, we must already have conceived of the idea of the structure itself; we cannot be discovering structures by conjecturing them.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
     A reaction: This has to be a crucial area of discussion. Do we have our heads full of abstract structures before we look out of the window? Externalism about the mind is important here; mind and world are not utterly distinct things.
Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
     Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model.
     From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
     A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator.
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
     Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out.
     From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
     A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra.
Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo]
     Full Idea: Structuralism can be distinguished from traditional Platonism in that it denies that mathematical objects from the same structure are ontologically independent of one another
     From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], III)
     A reaction: My instincts strongly cry out against all versions of this. If you are going to be a platonist (rather as if you are going to be religious) you might as well go for it big time and have independent objects, which will then dictate a structure.
'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
     Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4)
     A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns.