Combining Texts

All the ideas for 'Intro to G��del's Theorems', 'Contributions to Philosophy' and 'Principles of Philosophy of the Future'

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69 ideas

1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / b. Philosophy as transcendent
Only that which can be an object of religion is an object of philosophy [Feuerbach]
1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
Philosophy should not focus on names, but on the determined nature of things [Feuerbach]
1. Philosophy / H. Continental Philosophy / 1. Continental Philosophy
Modern philosophy begins with Descartes' abstraction from sensation and matter [Feuerbach]
Empiricism is right about ideas, but forgets man himself as one of our objects [Feuerbach]
2. Reason / B. Laws of Thought / 1. Laws of Thought
The laws of reality are also the laws of thought [Feuerbach]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
There cannot be a set theory which is complete [Smith,P]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order arithmetic can prove new sentences of first-order [Smith,P]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'partial function' maps only some elements to another set [Smith,P]
A 'total function' maps every element to one element in another set [Smith,P]
An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P]
The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
Two functions are the same if they have the same extension [Smith,P]
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
A 'natural deduction system' has no axioms but many rules [Smith,P]
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
No nice theory can define truth for its own language [Smith,P]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P]
A 'surjective' ('onto') function creates every element of the output set [Smith,P]
A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
5. Theory of Logic / K. Features of Logics / 3. Soundness
If everything that a theory proves is true, then it is 'sound' [Smith,P]
Soundness is true axioms and a truth-preserving proof system [Smith,P]
A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A theory is 'negation complete' if it proves all sentences or their negation [Smith,P]
'Complete' applies both to whole logics, and to theories within them [Smith,P]
A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P]
5. Theory of Logic / K. Features of Logics / 7. Decidability
'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P]
A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P]
Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P]
A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P]
A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P]
The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P]
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P]
5. Theory of Logic / K. Features of Logics / 9. Expressibility
Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
All numbers are related to zero by the ancestral of the successor relation [Smith,P]
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P]
Baby Arithmetic is complete, but not very expressive [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / c. Robinson arithmetic
Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P]
Robinson Arithmetic (Q) is not negation complete [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
7. Existence / A. Nature of Existence / 3. Being / a. Nature of Being
Absolute thought remains in another world from being [Feuerbach]
Being is what is undetermined, and hence indistinguishable [Feuerbach]
7. Existence / A. Nature of Existence / 3. Being / f. Primary being
Being posits essence, and my essence is my being [Feuerbach]
7. Existence / A. Nature of Existence / 3. Being / g. Particular being
Particularity belongs to being, whereas generality belongs to thought [Feuerbach]
7. Existence / A. Nature of Existence / 3. Being / h. Dasein (being human)
The only true being is of the senses, perception, feeling and love [Feuerbach]
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
Consciousness is absolute reality, and everything exists through consciousness [Feuerbach]
11. Knowledge Aims / C. Knowing Reality / 4. Solipsism
Ideas arise through communication, and reason is reached through community [Feuerbach]
12. Knowledge Sources / B. Perception / 6. Inference in Perception
In man the lowest senses of smell and taste elevate themselves to intellectual acts [Feuerbach]
18. Thought / E. Abstraction / 1. Abstract Thought
The new philosophy thinks of the concrete in a concrete (not a abstract) manner [Feuerbach]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / d. Biological ethics
Plotinus was ashamed to have a body [Feuerbach]
22. Metaethics / B. Value / 2. Values / g. Love
If you love nothing, it doesn't matter whether something exists or not [Feuerbach]
23. Ethics / F. Existentialism / 4. Boredom
Culture is now dominated by boredom, so universal it is unnoticed [Heidegger, by Aho]
24. Political Theory / A. Basis of a State / 1. A People / a. Human distinctiveness
Man is not a particular being, like animals, but a universal being [Feuerbach]
The essence of man is in community, but with distinct individuals [Feuerbach]
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
God's existence cannot be separated from essence and concept, which can only be thought as existing [Feuerbach]
28. God / C. Attitudes to God / 4. God Reflects Humanity
If God is only an object for man, then only the essence of man is revealed in God [Feuerbach]
God is what man would like to be [Feuerbach]
God is for us a mere empty idea, which we fill with our own ego and essence [Feuerbach]
29. Religion / B. Monotheistic Religion / 4. Christianity / a. Christianity
Catholicism concerns God in himself, Protestantism what God is for man [Feuerbach]
29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
Absolute idealism is the realized divine mind of Leibnizian theism [Feuerbach]