11 ideas
| 7661 | Truth is the opinion fated to be ultimately agreed by all investigators [Peirce] |
| Full Idea: The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real. | |
| From: Charles Sanders Peirce (How to Make our Ideas Clear [1878], p.38) | |
| A reaction: At least this affirms that truth is an ideal about which we dream, and is not confined merely to what we can actually know. But it rules out anything beyond the reach of all investigation, which seems a misconception of truth. What could angels know? |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 19089 | Our whole conception of an object is its possible practical consequences [Peirce] |
| Full Idea: Consider what effects, which might conceivably have practical bearings, we conceive the objects of our conceptions to have. Then, our conception of these effects is the whole of our conception of the object. | |
| From: Charles Sanders Peirce (How to Make our Ideas Clear [1878], EP i.132), quoted by Albert Atkin - Peirce 2 'early' | |
| A reaction: This is his 1878 version, which was fine-tuned later in life. He seems to have extended his principle to include possibilities, as well as the mere objects. That is, he moved beyond mere nominalism. |
| 7660 | We are aware of beliefs, they appease our doubts, and they are rules of action, or habits [Peirce] |
| Full Idea: A belief has just three properties: first, it is something that we are aware of; second, it appeases the irritation of doubt; and, third, it involves the establishment in our nature of a rule of action, or, say for short, a habit. | |
| From: Charles Sanders Peirce (How to Make our Ideas Clear [1878], p.28) | |
| A reaction: Peirce probably believed that Bismarck breathed oxygen, but was unaware of his belief, and no one ever dreamed of acting on such a belief, unless Bismarck was gasping for air. |
| 18278 | Kant showed that our perceptions are partly constructed from our concepts [Reichenbach] |
| Full Idea: It was Kant's great discovery that the object of knowledge is not simply given but constructed, and that it contains conceptual elements not contained in pure perception. | |
| From: Hans Reichenbach (The Theory of Relativity and A Priori Knowledge [1965], p.49), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap |
| 14906 | Non-positivist verificationism says only take a hypothesis seriously if it is scientifically based and testable [Ladyman/Ross on Peirce] |
| Full Idea: With Peirce, we endorse a non-positivist version of verificationism - no hypothesis should be taken seriously if apparently beyond our capacity to investigate, and serious metaphysics must concern at least two plausible scientific hypotheses. | |
| From: comment on Charles Sanders Peirce (How to Make our Ideas Clear [1878]) by J Ladyman / D Ross - Every Thing Must Go 1.3 | |
| A reaction: [compressed] They say this is NOT a theory about meaning, as 'The Big Bang was caused by Elvis' is perfectly meaningful. Verificationism always seems to rule out bold speculation. Don't say 'take string theory seriously', if we can't test it? |