24 ideas
| 13430 | Infinity: there is an infinity of distinguishable individuals [Ramsey] |
| Full Idea: The Axiom of Infinity means that there are an infinity of distinguishable individuals, which is an empirical proposition. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], §5) | |
| A reaction: The Axiom sounds absurd, as a part of a logical system, but Ramsey ends up defending it. Logical tautologies, which seem to be obviously true, are rendered absurd if they don't refer to any objects, and some of them refer to infinities of objects. |
| 13428 | Reducibility: to every non-elementary function there is an equivalent elementary function [Ramsey] |
| Full Idea: The Axiom of Reducibility asserted that to every non-elementary function there is an equivalent elementary function [note: two functions are equivalent when the same arguments render them both true or both false]. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], §2) | |
| A reaction: Ramsey in the business of showing that this axiom from Russell and Whitehead is not needed. He says that the axiom seems to be needed for induction and for Dedekind cuts. Since the cuts rest on it, and it is weak, Ramsey says it must go. |
| 13427 | Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey] |
| Full Idea: In 'a = b' either 'a' and 'b' are names of the same thing, in which case the proposition says nothing, or of different things, in which case it is absurd. In neither case is it an assertion of a fact; it only asserts when a or b are descriptions. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1) | |
| A reaction: This is essentially Frege's problem with Hesperus and Phosphorus. How can identities be informative? So 2+2=4 is extensionally vacuous, but informative because they are different descriptions. |
| 13334 | Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey] |
| Full Idea: Group A consists of contradictions which would occur in a logical or mathematical system, involving terms such as class or number. Group B contradictions are not purely logical, and contain some reference to thought, language or symbolism. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.171), quoted by Graham Priest - The Structure of Paradoxes of Self-Reference 1 | |
| A reaction: This has become the orthodox division of all paradoxes, but the division is challenged by Priest (Idea 13373). He suggests that we now realise (post-Tarski?) that language is more involved in logic and mathematics than we thought. |
| 13426 | Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey] |
| Full Idea: The formalists neglected the content altogether and made mathematics meaningless, but the logicians neglected the form and made mathematics consist of any true generalisations; only by taking account of both sides can we obtain an adequate theory. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1) | |
| A reaction: He says mathematics is 'tautological generalizations'. It is a criticism of modern structuralism that it overemphasises form, and fails to pay attention to the meaning of the concepts which stand at the 'nodes' of the structure. |
| 13425 | Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey] |
| Full Idea: The hopelessly inadequate formalist theory is, to some extent, the result of considering only the propositions of mathematics and neglecting the analysis of its concepts. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1) | |
| A reaction: You'll have to read Ramsey to see how this thought pans out, but it at least gives a pointer to how to go about addressing the question. |
| 8780 | Attributes are functions, not objects; this distinguishes 'square of 2' from 'double of 2' [Geach] |
| Full Idea: Attributes should not be thought of as identifiable objects. It is better to follow Frege and compare them to mathematical functions. 'Square of' and 'double of' x are distinct functions, even though they are not distinguishable in thought when x is 2. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §11) | |
| A reaction: Attributes are features of the world, of which animals are well aware, and the mathematical model is dubious when dealing with physical properties. The route to arriving at 2 is not the same concept as 2. There are many roads to Rome. |
| 11910 | Being 'the same' is meaningless, unless we specify 'the same X' [Geach] |
| Full Idea: "The same" is a fragmentary expression, and has no significance unless we say or mean "the same X", where X represents a general term. ...There is no such thing as being just 'the same'. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §16) | |
| A reaction: Geach seems oddly unaware of the perfect identity of Hespherus with Phosphorus. His critics don't spot that he was concerned with identity over time (of 'the same man', who ages). Perry's critique emphasises the type/token distinction. |
| 22328 | I just confront the evidence, and let it act on me [Ramsey] |
| Full Idea: I can but put the evidence before me, and let it act on my mind. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.202), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 70 'Deg' | |
| A reaction: Potter calls this observation 'downbeat', but I am an enthusiastic fan. It is roughly my view of both concept formation and of knowledge. You soak up the world, and respond appropriately. The trick is in the selection of evidence to confront. |
| 22325 | A belief is knowledge if it is true, certain and obtained by a reliable process [Ramsey] |
| Full Idea: I have always said that a belief was knowledge if it was 1) true, ii) certain, iii) obtained by a reliable process. | |
| From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.258), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 66 'Rel' | |
| A reaction: Not sure why it has to be 'certain' as well as 'true'. It seems that 'true' is objective, and 'certain' subjective. I think I know lots of things of which I am not fully certain. Reliabilism long preceded Alvin Goldman. |
| 8775 | A big flea is a small animal, so 'big' and 'small' cannot be acquired by abstraction [Geach] |
| Full Idea: A big flea or rat is a small animal, and a small elephant is a big animal, so there can be no question of ignoring the kind of thing to which 'big' or 'small' is referred and forming those concepts by abstraction. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §9) | |
| A reaction: Geach is attacking a caricature of the theory. Abstraction is a neat mental trick which has developed in stages, from big rats relative to us, to big relative to other rats, to the concept of 'relative' (Idea 8776!), to the concept of 'relative bigness'. |
| 8776 | We cannot learn relations by abstraction, because their converse must be learned too [Geach] |
| Full Idea: Abstractionists are unaware of the difficulty with relations - that they neither exist nor can be observed apart from the converse relation, the two being indivisible, as in grasping 'to the left of' and 'to the right of'. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §9) | |
| A reaction: It is hard to see how a rival account such as platonism could help. It seems obvious to me that 'right' and 'left' would be quite meaningless without some experience of things in space, including an orientation to them. |
| 2567 | You can't define real mental states in terms of behaviour that never happens [Geach] |
| Full Idea: We can't take a statement that two men, whose overt behaviour was not actually different, were in different states of mind as being really a statement that the behaviour of one man would have been different in hypothetical circumstances that never arose. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §3) | |
| A reaction: This is the whole problem with trying to define the mind as dispositions. The same might be said of properties, since some properties are active, but others are mere potential or disposition. Hence 'process' looks to me the most promising word for mind. |
| 2568 | Beliefs aren't tied to particular behaviours [Geach] |
| Full Idea: Is there any behaviour characteristic of a given belief? | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §4) | |
| A reaction: Well, yes. Belief that a dog is about to bite you. Belief that this nice food is yours, and you are hungry. But he has a good point. He is pointing out that the mental state is a very different thing from the 'disposition' to behave in a certain way. |
| 8781 | The mind does not lift concepts from experience; it creates them, and then applies them [Geach] |
| Full Idea: Having a concept is not recognizing a feature of experience; the mind makes concepts. We then fit our concepts to experience. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §11) | |
| A reaction: This seems to imply that we create concepts ex nihilo, which is a rather worse theory than saying that we abstract them from multiple (and multi-level) experiences. That minds create concepts is a truism. How do we do it? |
| 8769 | If someone has aphasia but can still play chess, they clearly have concepts [Geach] |
| Full Idea: If a man struck with aphasia can still play bridge or chess, I certainly wish to say he still has the concepts involved in the game, although he can no longer exercise them verbally. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §5) | |
| A reaction: Geach proceeds thereafter to concentrate on language, but this caveat is crucial. To suggest that concepts are entirely verbal has always struck me as ridiculous, and an insult to our inarticulate mammalian cousins. |
| 8770 | 'Abstractionism' is acquiring a concept by picking out one experience amongst a group [Geach] |
| Full Idea: I call 'abstractionism' the doctrine that a concept is acquired by a process of singling out in attention some one feature given in direct experience - abstracting it - and ignoring the other features simultaneously given - abstracting from them. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §6) | |
| A reaction: Locke seems to be the best known ancestor of this view, and Geach launches a vigorous attack against it. However, contemporary philosophers still refer to the process, and I think Geach should be crushed and this theory revived. |
| 8771 | 'Or' and 'not' are not to be found in the sensible world, or even in the world of inner experience [Geach] |
| Full Idea: Nowhere in the sensible world could you find anything to be suitably labelled 'or' or 'not'. So the abstractionist appeals to an 'inner sense', or hesitation for 'or', and of frustration or inhibition for 'not'. Personally I see a threat in 'or else'! | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §7) | |
| A reaction: This is a key argument of Geach's against abstractionism. As a logician he prefers to discuss connectives rather than, say, colours. I think they might be meta-abstractions, which you create internally once you have picked up the knack. |
| 8772 | We can't acquire number-concepts by extracting the number from the things being counted [Geach] |
| Full Idea: The number-concepts just cannot be got by concentrating on the number and abstracting from the kind of things being counted. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §8) | |
| A reaction: This point is from Frege - that if you 'abstract away' everything apart from the number, you are simply left with nothing in experience. The objection might, I think, be met by viewing it as second-order abstraction, perhaps getting to a pattern first. |
| 8773 | Abstractionists can't explain counting, because it must precede experience of objects [Geach] |
| Full Idea: The way counting is learned is wholly contrary to abstractionist preconceptions, because the series of numerals has to be learned before it can be applied. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §8) | |
| A reaction: You might learn to parrot the names of numbers, but you could hardly know what they meant if you couldn't count anything. See Idea 3907. I would have thought that individuating objects must logically and pedagogically precede counting. |
| 8774 | The numbers don't exist in nature, so they cannot have been abstracted from there into our languages [Geach] |
| Full Idea: The pattern of the numeral series that is grasped by a child exists nowhere in nature outside human languages, so the human race cannot possibly have discerned this pattern by abstracting it from some natural context. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §8) | |
| A reaction: This is a spectacular non sequitur, which begs the question. Abstractionists precisely claim that the process of abstraction brings numerals into human language from the natural context. Structuralism is an attempt to explain the process. |
| 8778 | Blind people can use colour words like 'red' perfectly intelligently [Geach] |
| Full Idea: It is not true that men born blind can form no colour-concepts; a man born blind can use the word 'red' with a considerable measure of intelligence; he can show a practical grasp of the logic of the word. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §10) | |
| A reaction: Weak. It is obvious that they pick up the word 'red' from the usage of sighted people, and the usage of the word doesn't guarantee a grasp of the concept, as when non-mathematicians refer to 'calculus'. Compare Idea 7377 and Idea 7866. |
| 8777 | If 'black' and 'cat' can be used in the absence of such objects, how can such usage be abstracted? [Geach] |
| Full Idea: Since we can use the terms 'black' and 'cat' in situations not including any black object or any cat, how could this part of the use be got by abstraction? | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §10) | |
| A reaction: [He is attacking H.H. Price] It doesn't seem a huge psychological leap to apply the word 'cat' when we remember a cat, and once it is in the mind we can play games with our abstractions. Cats are smaller than dogs. |
| 8779 | We can form two different abstract concepts that apply to a single unified experience [Geach] |
| Full Idea: It is impossible to form the concept of 'chromatic colour' by discriminative attention to a feature given in my visual experience. In seeing a red window-pane, I do not have two sensations, one of redness and one of chromatic colour. | |
| From: Peter Geach (Mental Acts: their content and their objects [1957], §10) | |
| A reaction: Again Geach begs the question, because abstractionists claim that you can focus on two different 'aspects' of the one experience, as that it is a 'window', or it is 'red', or it is not a wall, or it is not monochrome. |