9 ideas
10185 | Set theory is the standard background for modern mathematics [Burgess] |
Full Idea: In present-day mathematics, it is set theory that serves as the background theory in which other branches of mathematics are developed. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: [He cites Bourbaki as an authority for this] See Benacerraf for a famous difficulty here, when you actually try to derive an ontology from the mathematicians' working practices. |
10184 | Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess] |
Full Idea: On the structuralist interpretation, theorems of analysis concerning the real numbers R are about all complete ordered fields. So R, which appears to be the name of a specific structure, is taken to be a variable ranging over structures. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: Since I am beginning to think that nearly all linguistic expressions should be understood as variables, I find this very appealing, even if Burgess hates it. Terms slide and drift, and are vague, between variable and determinate reference. |
10189 | There is no one relation for the real number 2, as relations differ in different models [Burgess] |
Full Idea: One might meet the 'Van Inwagen Problem' by saying that the intrinsic properties of the object playing the role of 2 will differ from one model to another, so that no statement about the intrinsic properties of 'the' real numbers will make sense. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
A reaction: There seems to be a potential confusion among opponents of structuralism between relations at the level of actual mathematical operations, and generalisations about relations, which are captured in the word 'patterns'. Call them 'meta-relations'? |
10186 | 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. |
10187 | 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. |
10188 | 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). |
14380 | The distinction between necessary and essential properties can be ignored [Rocca] |
Full Idea: Some philosophers distinguish between necessary properties and essential properties. This distinction is irrelevant to my purposes. Following Yablo, I shall ignore this distinction in what follows. | |
From: Michael della Rocca (Essentialists and Essentialism [1996], I n1) | |
A reaction: This is two years after Kit Fine's seminal paper suggesting the distinction is real. The first step towards a good metaphysics is to realise that Della Rocca and Yablo have made a horrible mistake. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |