31 ideas
| 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) |
| 9837 | 0 is not a number, as it answers 'how many?' negatively [Husserl, by Dummett] |
| Full Idea: Husserl contends that 0 is not a number, on the grounds that 'nought' is a negative answer to the question 'how many?'. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.144) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: I seem to be in a tiny minority in thinking that Husserl may have a good point. One apple is different from one orange, but no apples are the same as no oranges. That makes 0 a very peculiar number. See Idea 9838. |
| 9576 | Multiplicity in general is just one and one and one, etc. [Husserl] |
| Full Idea: Multiplicity in general is no more than something and something and something, etc.; ..or more briefly, one and one and one, etc. | |
| From: Edmund Husserl (Philosophy of Arithmetic [1894], p.85), quoted by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic' | |
| A reaction: Frege goes on to attack this idea fairly convincingly. It seems obvious that it is hard to say that you have seventeen items, if the only numberical concept in your possession is 'one'. How would you distinguish 17 from 16? What makes the ones 'multiple'? |
| 17444 | Husserl said counting is more basic than Frege's one-one correspondence [Husserl, by Heck] |
| Full Idea: Husserl famously argued that one should not explain number in terms of equinumerosity (or one-one correspondence), but should explain equinumerosity in terms of sameness of number, which should be characterised in terms of counting. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: [Heck admits he hasn't read the Husserl] I'm very sympathetic to Husserl, though nearly all modern thinking favours Frege. Counting connects numbers to their roots in the world. Mathematicians seem oblivious of such things. |
| 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) |
| 6472 | Continuity is a sufficient criterion for the identity of a rock, but not for part of a smooth fluid [Russell] |
| Full Idea: Continuity is not a sufficient criterion of material identity; it is sufficient in many cases, such as rocks and tables, where the appearances change slowly, but in others, such as the parts of an approximately homogeneous fluid, it fails us utterly. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: It might be debatable to what extent the 'parts' of a homogeneous fluid have identity. How many 'parts' are there in a glass of water? This seems, now, a problem for internalists; externalists can define the identity by the unseen molecules. |
| 6473 | Physical things are series of appearances whose matter obeys physical laws [Russell] |
| Full Idea: We may lay down the following definition: Physical things are those series of appearances whose matter obeys the laws of physics. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: We will then have to define the laws of physic without making any reference to 'physical things'. There is an obvious suspicion of circularity somewhere here. I find it very odd to define objects just in terms of their appearances. |
| 6465 | We need not deny substance, but there seems no reason to assert it [Russell] |
| Full Idea: It is not necessary to deny a substance or substratum underlying appearances; it is merely expedient (by the application of Occam's Razor) to abstain from asserting this unnecessary entity. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §V) | |
| A reaction: Russell then goes on to struggle heroically in attempts to give accounts of 'matter' and 'objects' entirely in terms of 'sense-data'. If he failed, as many think he did, should we go back to belief in Aristotelian substance? |
| 6471 | The assumption by physicists of permanent substance is not metaphysically legitimate [Russell] |
| Full Idea: The assumption of permanent substance, which technically underlies the procedure of physics, cannot of course be regarded as metaphysically legitimate. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §XI) | |
| A reaction: It is a moot point whether physicists still thought this way after the full arrival of quantum theory in 1926. Russell raises all sorts of nice questions about the relationship between physics and philosophy here. I'm on Russell's side. |
| 6466 | Where possible, logical constructions are to be substituted for inferred entities [Russell] |
| Full Idea: The supreme maxim in scientific philosophising is this: Wherever possible, logical constructions are to be substituted for inferred entities. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VI) | |
| A reaction: This seems to represent Russell's first move (in 1914) into what looks like phenomenalism. One might ask what is the difference between 'logical constructions' and 'inferred entities'. The latter appear to have unity, so I prefer them. |
| 6483 | Russell held that we are aware of states of our own brain [Russell, by Robinson,H] |
| Full Idea: Russell held that we are aware of states of our own brain. | |
| From: report of Bertrand Russell (The Relation of Sense-Data to Physics [1914]) by Howard Robinson - Perception 1.1 | |
| A reaction: I can't say that I had ever intepreted Russell in this way, but it is a wonderfully thought-provoking idea. All the time that I thought I was looking at a table, I was just looking at my own brain, and drawing an unspoken inference that a table caused it. |
| 8244 | Sense-data are qualities devoid of subjectivity, which are the basis of science [Russell, by Deleuze/Guattari] |
| Full Idea: Rather than oppose sensory knowledge and scientific knowledge, we should identify the sensibilia that are peculiar to science. This is what Russell did when he evoked sense-data, qualities devoid of all subjectivity. | |
| From: report of Bertrand Russell (The Relation of Sense-Data to Physics [1914]) by G Deleuze / F Guattari - What is Philosophy? 2.5 | |
| A reaction: An interesting observation. Russell is striking for his lack of interest in theories of arts and ethics, and his whole work focuses on understanding the scientific view. What is involved in sensibilia is a key modern issue (e.g. McDowell). |
| 6462 | Sense-data are not mental, but are part of the subject-matter of physics [Russell] |
| Full Idea: I regard sense-data as not mental, and as being, in fact, part of the actual subject-matter of physics. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: Russell had clearly given himself an ontological problem with the introduction of sense-data, and this is his drastic solution. In 1912 his account seems ambiguous between sense-data being mental and being physical. |
| 6463 | Sense-data are objects, and do not contain the subject as part, the way beliefs do [Russell] |
| Full Idea: Logically a sense-datum is an object, a particular of which the subject is aware; it does not contain the subject as a part, as for example beliefs and volitions do. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IV) | |
| A reaction: This very firmly rejects any notion that a sense-datum is mental. It is a left as a strange sort of object which gets as close as it is possible to get to the 'borders' of the mind, without actually becoming part of it. |
| 6464 | Sense-data are usually objects within the body, but are not part of the subject [Russell] |
| Full Idea: The sense-datum is an external object of which in sensation the subject is aware; it is true that the sense-datum is in many cases in the subject's body, but the subject's body is as distinct from the subject as tables and chairs are. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IV) | |
| A reaction: This is probably Russell's clearest statement of the nature of sense-data, which are objects within the subjects body, but are not part of the mind. So once again we come up against the question of their ontology. Are they made of neurons? |
| 6467 | No sensibile is ever a datum to two people at once [Russell] |
| Full Idea: No sensibile is ever a datum to two people at once. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VII) | |
| A reaction: So a loud bang has to broken down into an almost infinite number of sound sensibilia - each one presumably the size of the apperture of a small ear. This is beginning to sound a bit silly. |
| 6459 | We do not know whether sense-data exist as objects when they are not data [Russell] |
| Full Idea: We do not know, except by means of more or less precarious inferences, whether the objects which are at one time sense-data continue to exist at times when they are not data. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §II) | |
| A reaction: Note that he actually refers to sense-data as 'objects'. It shows how thoroughly reified they are in his theory if they have the possibility of independent existence. This invites the question 'what are they made of?' |
| 6461 | Ungiven sense-data can no more exist than unmarried husbands [Russell] |
| Full Idea: We cannot ask, 'Can sense-data exist without being given?' for that is like asking, 'Can husbands exist without being married?' | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: This follows hard on Idea 6460, which introduces the idea of 'sensibilia' for things which are like sense-data, but are not 'given'. This is a new distinction in 1914, which he had not made in 1912. |
| 6460 | 'Sensibilia' are identical to sense-data, without actually being data for any mind [Russell] |
| Full Idea: I shall give the name 'sensibilia' to those objects which have the same metaphysical and physical status as sense-data without necessarily being data to any mind. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §III) | |
| A reaction: This is his response to the problem of whether sense-data can exist independently of experience, which was unclear in 1912. Presumably sensibilia are objects which are possible sources of experience, but that seems to cover most objects. |
| 6458 | Individuating sense-data is difficult, because they divide when closely attended to [Russell] |
| Full Idea: There is some difficulty in deciding what is to be considered one sense-datum: often attention causes divisions to appear where, so far as can be discovered, there were no divisions before. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §II) | |
| A reaction: This was, I suspect, why Russell had dropped the idea of sense-data by 1921. He does, however, say that they are the last unit in analysis, rather than being the most basic unit of perception. In other words, they are purely theoretical. |
| 6469 | Sense-data may be subjective, if closing our eyes can change them [Russell] |
| Full Idea: One reason often alleged for the subjectivity of sense-data is that the appearance of a thing itself may change when we find it hard to suppose that the thing itself has changed - as when we shut our eyes, or screw them up to make things look double. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VIII) | |
| A reaction: Russell firmly denies that they are subjective. These examples are also said to support to proposed existence of sense-data in the first place, since they show the gap between appearance and reality. |
| 9575 | Husserl identifies a positive mental act of unification, and a negative mental act for differences [Husserl, by Frege] |
| Full Idea: Husserl identifies a 'unitary mental act' where several contents are connected or related to one another, and also a difference-relation where two contents are related to one another by a negative judgement. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.73-74) by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic' p.322 | |
| A reaction: Frege is setting this up ready for a fairly vicious attack. Where Hume has a faculty for spotting resemblances, it is not implausible that we should also be hard-wired to spot differences. 'You look different; have you changed your hair style?' |
| 21214 | We clarify concepts (e.g. numbers) by determining their psychological origin [Husserl, by Velarde-Mayol] |
| Full Idea: Husserl said that the clarification of any concept is made by determining its psychological origin. He is concerned with the psychological origins of the operation of calculating cardinal numbers. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Victor Velarde-Mayol - On Husserl 2.2 | |
| A reaction: This may not be the same as the 'psychologism' that Frege so despised, because Husserl is offering a clarification, rather than the intrinsic nature of number concepts. It is not a theory of the origin of numbers. |
| 9819 | Psychologism blunders in focusing on concept-formation instead of delineating the concepts [Dummett on Husserl] |
| Full Idea: Husserl substitutes his account of the process of concept-formation for a delineation of the concept. It is above all in making this substitution that psychologism is objectionable (and Frege opposed it so vehemently). | |
| From: comment on Edmund Husserl (Philosophy of Arithmetic [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.2 | |
| A reaction: While this is a powerful point which is a modern orthodoxy, it hardly excludes a study of concept-formation from being of great interest for other reasons. It may not appeal to logicians, but it is crucial part of the metaphysics of nature. |
| 9851 | Husserl wanted to keep a shadowy remnant of abstracted objects, to correlate them [Dummett on Husserl] |
| Full Idea: Husserl saw that abstracted units, though featureless, must in some way retain their distinctness, some shadowy remnant of their objects. So he wanted to correlate like-numbered sets, not just register their identity, but then abstractionism fails. | |
| From: comment on Edmund Husserl (Philosophy of Arithmetic [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.12 | |
| A reaction: Abstractionism is held to be between the devil and the deep blue sea, of depending on units which are identifiable, when they are defined as devoid of all individuality. We seem forced to say that the only distinction between them is countability. |
| 6470 | Matter is the limit of appearances as distance from the object diminishes [Russell] |
| Full Idea: We offer the following tentative definition: The matter of a given thing is the limit of its appearances as their distance from the thing diminishes. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §IX) | |
| A reaction: This strikes me as empiricism gone mad. Russell is famous for being a 'realist', but you would hardly know it at this point. Personally I put emphasis on 'best explanation', which fairly simply delivers most of our commonsense understandings of reality. |
| 6468 | There is 'private space', and there is also the 'space of perspectives' [Russell] |
| Full Idea: In addition to the private spaces, ..there is the 'space of perspectives', since each private world may be regarded as the appearance which the universe presents from a certain point of view. | |
| From: Bertrand Russell (The Relation of Sense-Data to Physics [1914], §VII) | |
| A reaction: This replaces his concept of 'public space', which he introduced in 1912. Russell gradually dropped this, but I like the idea that we somehow directly perceive space in two ways simultaneously (which led him to say that space is six-dimensional). |