Combining Philosophers

All the ideas for Peter B. Lewis, Michael Potter and Charles Taylor

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25 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter]
     Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning.
     From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1)
     A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Usually the only reason given for accepting the empty set is convenience [Potter]
     Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinity: There is at least one limit level [Potter]
     Full Idea: Axiom of Infinity: There is at least one limit level.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9)
     A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
Nowadays we derive our conception of collections from the dependence between them [Potter]
     Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2)
     A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?'
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter]
     Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5)
     A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous.
4. Formal Logic / G. Formal Mereology / 1. Mereology
Mereology elides the distinction between the cards in a pack and the suits [Potter]
     Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1)
     A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled.
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
We can formalize second-order formation rules, but not inference rules [Potter]
     Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2)
     A reaction: He cites Gödel's First Incompleteness theorem for this.
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter]
     Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1)
     A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth).
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
     Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
     Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms).
     From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2)
     A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying.
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
A relation is a set consisting entirely of ordered pairs [Potter]
     Full Idea: A set is called a 'relation' if every element of it is an ordered pair.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7)
     A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related.
9. Objects / B. Unity of Objects / 2. Substance / b. Need for substance
If dependence is well-founded, with no infinite backward chains, this implies substances [Potter]
     Full Idea: The argument that the relation of dependence is well-founded a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
     A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here.
9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
Collections have fixed members, but fusions can be carved in innumerable ways [Potter]
     Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1)
     A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go.
10. Modality / A. Necessity / 1. Types of Modality
Priority is a modality, arising from collections and members [Potter]
     Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
     A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background.
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / d. Absolute idealism
Fichte, Schelling and Hegel rejected transcendental idealism [Lewis,PB]
     Full Idea: Fichte, Schelling and Hegel were united in their opposition to Kant's Transcendental Idealism.
     From: Peter B. Lewis (Schopenhauer [2012], 3)
     A reaction: That is, they preferred genuine idealism, to the mere idealist attitude Kant felt that we are forced to adopt.
Fichte, Hegel and Schelling developed versions of Absolute Idealism [Lewis,PB]
     Full Idea: At the University of Jena, Fichte, Hegel and Schelling critically developed aspects of Kant's philosophy, each in his own way, thereby giving rise to the movement known as Absolute Idealism, see reality as universal God-like self-consciousness.
     From: Peter B. Lewis (Schopenhauer [2012], 2)
     A reaction: Is asking how anyone can possibly have believed such a bizarre and ridiculous idea a) uneducated, b) stupid, c) unimaginative, or d) very sensible? It sounds awfully like Spinoza's concept of God. Also Anaxagoras.
16. Persons / A. Concept of a Person / 4. Persons as Agents
The modern self has disengaged reason, self-exploration, and personal commitment [Taylor,C]
     Full Idea: The modern notion of the self is defined by disengaged reason (with its associated freedom and dignity), by self-exploration, and by personal commitment.
     From: Charles Taylor (Sources of the Self [1989], §13.1)
     A reaction: Taylor makes a good case that this broader view of how the self is seen is as important as narrow debates about personal identity.
16. Persons / B. Nature of the Self / 2. Ethical Self
My aim is to map the connections between our sense of self and our moral understanding [Taylor,C]
     Full Idea: My entire way of proceeding involves mapping connections between the sense of the self and moral visions, between identity and the good.
     From: Charles Taylor (Sources of the Self [1989], Pref)
     A reaction: An interesting project. Modern brain research supports the idea that emotions and values are tightly integrated into al thought.
16. Persons / E. Rejecting the Self / 3. Narrative Self
I can only be aware of myself as a person who changes by means of my personal history [Taylor,C]
     Full Idea: As a being who grows and becomes I can only know myself through the history of my maturations and regressions, overcomings and defeats.
     From: Charles Taylor (Sources of the Self [1989], §2.3)
     A reaction: An important insight. My immediate sense of self makes my personal history central, not an extra. But a history must be a history OF something.
22. Metaethics / C. Ethics Foundations / 1. Nature of Ethics / b. Defining ethics
Selfhood and moral values are inextricably intertwined [Taylor,C]
     Full Idea: Selfhood and the good, or in another way selfhood and morality, turn out to be inextricably intertwined.
     From: Charles Taylor (Sources of the Self [1989], §1.1)
     A reaction: This seems an inevitable convergence of three centuries of thought about personal identity and morality.
23. Ethics / C. Virtue Theory / 3. Virtues / e. Honour
Willingness to risk life was the constitutive quality of the man of honour [Taylor,C]
     Full Idea: Willingness to risk life was the constitutive quality of the man of honour.
     From: Charles Taylor (Sources of the Self [1989], §13.1)
     A reaction: Which is why war is required. The growth of civil society meant the inevitable rise of other virtues.
23. Ethics / C. Virtue Theory / 3. Virtues / h. Respect
To have respect for people, you must feel their claims, or their injustices, or hold them in awe [Taylor,C]
     Full Idea: If you want to discriminate more finely what makes humans worthy of respect, you must call to mind the claim of human suffering, or what is repugnant about justice, or the awe you feel about human life.
     From: Charles Taylor (Sources of the Self [1989], §1.1)
     A reaction: A persuasive part of the claim that such feelings are inseparable from thinking about people in any way at all.
23. Ethics / D. Deontological Ethics / 4. Categorical Imperative
Consistency presupposes intrinsic description [Taylor,C]
     Full Idea: The issue of consistency presupposes intrinsic description.
     From: Charles Taylor (Sources of the Self [1989], §1.1)
     A reaction: This may be the key criticism of Kant. The so-called 'maxim' of an action can be almost infinitely re-expressed to suit the agent.
23. Ethics / E. Utilitarianism / 1. Utilitarianism
In later utilitarianism the modern stress on freedom leads to the rejection of paternalism [Taylor,C]
     Full Idea: In mature utilitarianism , the stress on modern freedom emerges in the rejection of paternalism.
     From: Charles Taylor (Sources of the Self [1989], §3.3)
     A reaction: This seems good; it is the beginnings of a rejection of paternalism. What is better, happiness or freedom? What is the value of freedom?
28. God / A. Divine Nature / 6. Divine Morality / d. God decrees morality
Nominalists defended the sovereignty of God against the idea of natural existing good and evil [Taylor,C]
     Full Idea: Late medieval nominalism defended the sovereignty of God as incompatible with there being an order in nature which by itself defined good and bad.
     From: Charles Taylor (Sources of the Self [1989], §3.3)
     A reaction: Part of their attack on Platonism. But what made them place such a high value on the sovereignty of God?