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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique

[objections to structuralism about mathematics]

23 ideas
If numbers are supposed to be patterns, each number can have many patterns [Frege]
     Full Idea: Patterns can be completely different while the number of their elements remains the same, so that here we would have different distinct fives, sixes and so forth.
     From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §41)
     A reaction: A blow to my enthusiasm for Michael Resnik's account of maths as patterns. See, for example, Ideas 6296 and 6301. We are clearly set up to spot patterns long before we arrive at the abstract concepts of numbers. We see the same number in two patterns.
Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
     Full Idea: It is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as constitute a progression. If they are anything at all, they must be intrinsically something.
     From: Bertrand Russell (The Principles of Mathematics [1903], §242)
     A reaction: This is the obvious platonist response to the incipient doctrine of structuralism. We have a chicken-and-egg problem. Bricks need intrinsic properties to make a structure. A structure isomorphic to numbers is not thereby the numbers.
Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
     Full Idea: In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures.
     From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4
     A reaction: This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another.
The identity of a number may be fixed by something outside structure - by counting [Dummett]
     Full Idea: The identity of a mathematical object may sometimes be fixed by its relation to what lies outside the structure to which it belongs. It is more fundamental to '3' that if certain objects are counted, there are three of them.
     From: Michael Dummett (Frege philosophy of mathematics [1991], Ch. 5)
     A reaction: This strikes me as Dummett being pushed (by his dislike of the purely abstract picture given by structuralism) back to a rather empiricist and physical view of numbers, though he would totally deny that.
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett]
     Full Idea: The number 0 is not differentiated from 1 by its position in a progression, otherwise there would be no difference between starting with 0 and starting with 1. That is enough to show that numbers are not identifiable just as positions in structures.
     From: Michael Dummett (Frege philosophy of mathematics [1991], Ch. 5)
     A reaction: This sounds conclusive, but doesn't feel right. If numbers are a structure, then where you 'start' seems unimportant. Where do you 'start' in St Paul's Cathedral? Starting sounds like a constructivist concept for number theory.
The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
     Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1)
     A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done.
Numbers can't be positions, if nothing decides what position a given number has [Bostock]
     Full Idea: There is no ground for saying that a number IS a position, if the truth is that there is nothing to determine which number is which position.
     From: David Bostock (Philosophy of Mathematics [2009], 6.4)
     A reaction: If numbers lose touch with the empirical ability to count physical objects, they drift off into a mad world where they crumble away.
Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
     Full Idea: Structuralism begins from a false premise, namely that numbers have no properties other than their relations to other numbers.
     From: David Bostock (Philosophy of Mathematics [2009], 6.5)
     A reaction: Well said. Describing anything purely relationally strikes me as doomed, because you have to say why those things relate in those ways.
We don't need 'abstract structures' to have structural truths about successor functions [Lewis]
     Full Idea: We needn't believe in 'abstract structures' to have general structural truths about all successor functions.
     From: David Lewis (Mathematics is Megethology [1993], p.16)
If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
     Full Idea: The relativization of ontology to theory in structuralism can't avoid carrying with it a relativization of truth-value, which would compromise the objectivity which structuralists wish to claim for mathematics.
     From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26)
     A reaction: This is the attraction of structures which grow out of the physical world, where truth-value is presumably not in dispute.
The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
     Full Idea: It is not clear how the view that natural numbers are purely intra-structural 'objects' can be squared with the widespread use of numerals outside purely arithmetical contexts.
     From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26)
     A reaction: I don't understand this objection. If they refer to quantity, they are implicitly cardinal. If they name things in a sequence they are implicitly ordinal. All users of numbers have a grasp of the basic structure.
How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil]
     Full Idea: I do not understand how structures could serve as truthmakers for mathematical truths, ...Mathematical truths are not true in virtue of any way the universe is. ...Mathematical truths hold, whatever ways the universe is.
     From: John Heil (The Universe as We Find It [2012], 08.08)
     A reaction: I like the idea of enquiring about truthmakers for mathematical truths (and my view is more empirical than Heil's), but I think it may be a misunderstanding to think that structures are intended as truthmakers. Mathematics just IS structures?
Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
     Full Idea: According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)
     A reaction: Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones.
If set theory is used to define 'structure', we can't define set theory structurally [Burgess]
     Full Idea: It is to set theory that one turns for the very definition of 'structure', ...and this creates a problem of circularity if we try to impose a structuralist interpretation on set theory.
     From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1)
     A reaction: This seems like a nice difficulty, especially if, like Shapiro, you wade in and try to give a formal account of structures and patterns. Resnik is more circumspect and vague.
Abstract algebra concerns relations between models, not common features of all the models [Burgess]
     Full Idea: Abstract algebra, such as group theory, is not concerned with the features common to all models of the axioms, but rather with the relationships among different models of those axioms (especially homomorphic relation functions).
     From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1)
     A reaction: It doesn't seem to follow that structuralism can't be about the relations (or patterns) found when abstracting away and overviewing all the models. One can study family relations, or one can study kinship in general.
How can mathematical relations be either internal, or external, or intrinsic? [Burgess]
     Full Idea: The 'Van Inwagen Problem' for structuralism is of explaining how a mathematical relation (such as set membership, or the ratios of an ellipse) can fit into one of the three scholastics types of relations: are they internal, external, or intrinsic?
     From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5)
     A reaction: The difficulty is that mathematical objects seem to need intrinsic properties to get any of these three versions off the ground (which was Russell's complaint against structures).
Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR]
     Full Idea: Set theory is at the very heart of mathematics; it may even be all there is to mathematics. The notion of set, however, seems quite contrary to the spirit of structuralism.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
     A reaction: So much the worse for sets, I say. You can, for example, define ordinality in terms of sets, but that is no good if ordinality is basic to the nature of numbers, rather than a later addition.
The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
     Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference.
     From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
     A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity.
For mathematical objects to be positions, positions themselves must exist first [MacBride]
     Full Idea: The identification of mathematical objects with positions in structures rests upon the prior credibility of the thesis that positions are objects in their own right.
     From: Fraser MacBride (Structuralism Reconsidered [2007], §3)
     A reaction: Sounds devastating, but something has to get the whole thing off the ground. This is why Resnik's word 'patterns' is so appealing. Patterns stare you in the face, and they don't change if all the objects making it up are replaced by others.
Structuralism is right about algebra, but wrong about sets [Linnebo]
     Full Idea: Against extreme views that all mathematical objects depend on the structures to which they belong, or that none do, I defend a compromise view, that structuralists are right about algebraic objects (roughly), but anti-structuralists are right about sets.
     From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], Intro)
In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo]
     Full Idea: If objects depend on the other objects, this would mean an 'upward' dependence, in that they depend on the structure to which they belong, where the physical realm has a 'downward' dependence, with structures depending on their constituents.
     From: Øystein Linnebo (Structuralism and the Notion of Dependence [2008], III)
     A reaction: This nicely captures an intuition I have that there is something wrong with a commitment primarily to 'structures'. Our only conception of such things is as built up out of components. Not that I am committing to mathematical 'components'!
Some questions concern mathematical entities, rather than whole structures [Koslicki]
     Full Idea: Those who hold that not all mathematical questions can be concerned with structural matters can point to 'why are π or e transcendental?' or 'how are the prime numbers distributed?' as questions about particular features in the domain.
     From: Kathrin Koslicki (The Structure of Objects [2008], 9.3.1 n6)
     A reaction: [She cites Mac Lane on this] The reply would have to be that we only have those particular notions because we have abstracted them from structures, as in deriving π for circles.
If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]
     Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2)
     A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend.