31 ideas
| 1848 | We are coerced into assent to a truth by reason's violence [Aquinas] |
| Full Idea: We are coerced into assent to a truth by reason's violence. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.10) |
| 1858 | The mind is compelled by necessary truths, but not by contingent truths [Aquinas] |
| Full Idea: Mind is compelled by necessary truths that can't be regarded as false, but not by contingent ones that might be false. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 12) |
| 1852 | For the mind Good is one truth among many, and Truth is one good among many [Aquinas] |
| Full Idea: Good itself as taken in by mind is one truth among others, and truth itself as goal of mind's activity is one good among others. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 1860 | Knowledge may be based on senses, but we needn't sense all our knowledge [Aquinas] |
| Full Idea: All our knowledge comes through our senses, but that doesn't mean that everything we know is sensed. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 18) |
| 1855 | If we saw something as totally and utterly good, we would be compelled to will it [Aquinas] |
| Full Idea: Something apprehended to be good and appropriate in any and every circumstance that could be thought of would compel us to will it. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) |
| 1861 | The will is not compelled to move, even if pleasant things are set before it [Aquinas] |
| Full Idea: The will is not compelled to move, for it doesn't have to want the pleasant things set before it. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 21) |
| 1853 | Because the will moves by examining alternatives, it doesn't compel itself to will [Aquinas] |
| Full Idea: Because will moves itself by deliberation - a kind of investigation which doesn't prove some one way correct but examines the alternatives - will doesn't compel itself to will. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) |
| 1856 | Nothing can be willed except what is good, but good is very varied, and so choices are unpredictable [Aquinas] |
| Full Idea: Nothing can be willed except good, but many and various things are good, and you can't conclude from this that wills are compelled to choose this or that one. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 05) |
| 1849 | Since will is a reasoning power, it can entertain opposites, so it is not compelled to embrace one of them [Aquinas] |
| Full Idea: Reasoning powers can entertain opposite objects. Now will is a reasoning power, so will can entertain opposites and is not compelled to embrace one of them. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.x2) |
| 1862 | However habituated you are, given time to ponder you can go against a habit [Aquinas] |
| Full Idea: However habituated you are, given time to ponder you can go against a habit. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 24) |
| 1854 | We must admit that when the will is not willing something, the first movement to will must come from outside the will [Aquinas] |
| Full Idea: We are forced to admit that, in any will that is not always willing, the very first movement to will must come from outside, stimulating the will to start willing. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) | |
| A reaction: cf Nietzsche |
| 1847 | The will must aim at happiness, but can choose the means [Aquinas] |
| Full Idea: The will is compelled by its ultimate goal (to achieve happiness), but not by the means to achieve it. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.07) |
| 1857 | We don't have to will even perfect good, because we can choose not to think of it [Aquinas] |
| Full Idea: The will can avoid actually willing something by avoiding thinking of it, since mental activity is subject to will. In this respect we aren't compelled to will even total happiness, which is the only perfect good. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 07) |
| 1846 | The will can only want what it thinks is good [Aquinas] |
| Full Idea: Will's object is what is good, and so it cannot will anything but what is good. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.06) |
| 1850 | Without free will not only is ethical action meaningless, but also planning, commanding, praising and blaming [Aquinas] |
| Full Idea: If we are not free to will in any way, but are compelled, everything that makes up ethics vanishes: pondering action, exhorting, commanding, punishing, praising, condemning. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) | |
| A reaction: If doesn't require some magical 'free will' to avoid compulsions. All that is needed is freedom to enact your own willing, rather than someone else's. |
| 1851 | Good applies to goals, just as truth applies to ideas in the mind [Aquinas] |
| Full Idea: Good applies to all goals, just as truth applies to all forms mind takes in. | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.reply) | |
| A reaction: In danger of being tautological, if good is understood as no more than the goal of actions. It seems perfectly possibly to pursue a wicked end, and perhaps feel guilty about it. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 1859 | Even a sufficient cause doesn't compel its effect, because interference could interrupt the process [Aquinas] |
| Full Idea: Even a sufficient cause doesn't always compel its effect, since it can sometimes be interfered with so that its effect doesn't happen | |
| From: Thomas Aquinas (Quaestiones Disputatae de Malo [1271], Q6.h to 15) |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |