16 ideas
| 19441 | All philosophies presuppose their historical moment, and arise from it [Feuerbach] |
| Full Idea: Every philosophy originates as a manifestation of its time; its origin presupposes its historical time. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.59) | |
| A reaction: There seems to be widespread agreement among continental philosophers about this idea, whereas analytic philosophers largely ignore, and treat Plato as if he were a current professor in Chicago. |
| 19442 | I don't study Plato for his own sake; the primary aim is always understanding [Feuerbach] |
| Full Idea: Plato in writing is only a means for me; that which is primary and a priori, that which is the ground to which all is ultimately referred, is understanding. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.63) | |
| A reaction: It always seems to that the main aim of philosophy is understanding - which is why its central activity is explanation. |
| 19444 | Each proposition has an antithesis, and truth exists as its refutation [Feuerbach] |
| Full Idea: Every intellectual determination has its antithesis, its contradiction. Truth exists not in unity with, but in refutation of its opposite. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.72) | |
| A reaction: This appears to be a rejection of the 'synthesis' in Hegel, in favour of what strikes me as a rather more sensible interpretation of the modern dialectic. Being exists in contrast to nothingness, and truth exists in contrast to its negation? |
| 19445 | A dialectician has to be his own opponent [Feuerbach] |
| Full Idea: A thinker is a dialectician only insofar as he is his own opponent. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.72) | |
| A reaction: Quite an inspirational slogan for beginners in philosophy. How many non-philosophers are willing to be their own opponent. In law courts and the House of Commons we assign the roles to separate persons. Hence rhetoric replaces reason? |
| 19443 | Truth forges an impersonal unity between people [Feuerbach] |
| Full Idea: The urge to communicate is a fundamental urge - the urge for truth. ...That which is true belongs neither to me nor exclusively to you, but is common to all. The thought in which 'I' and 'You' are united is a true thought. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.65) | |
| A reaction: Sceptics may doubt that there are such truths, but this is certainly how we experience agreement - that there is some truth shared between us which is no longer the possession of either of us. Nice idea. |
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 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). |
| 19446 | To our consciousness it is language which looks unreal [Feuerbach] |
| Full Idea: To sensuous consciousness it is precisely language that is unreal, nothing. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.77) | |
| A reaction: Offered as a corrective to the view that our ontological commitments entirely concern what we are willing to say. |
| 19447 | The Absolute is the 'and' which unites 'spirit and nature' [Feuerbach] |
| Full Idea: The Absolute is spirit and nature. ...But what then is the Absolute? Nothing other than this 'and', that is, the unity of spirit and nature. | |
| From: Ludwig Feuerbach (Towards a Critique of Hegel's Philosophy [1839], p.82) | |
| A reaction: This is Feuerbach's spin on Hegel. He has been outlining idealist philosophy and the philosophy of nature in Schelling. Is this Spinoza's one substance? |