26 ideas
16877 | A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege] |
Full Idea: We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court. It contrasts with an 'analytic' definition. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.210) | |
A reaction: An analytic definition is evidently a deconstruction of a past constructive definition. Fregean definition is a creative activity. |
11219 | Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta] |
Full Idea: Frege has defended the austere view that, in mathematics at least, only stipulative definitions should be countenanced. | |
From: report of Gottlob Frege (Logic in Mathematics [1914]) by Anil Gupta - Definitions 1.3 | |
A reaction: This sounds intriguingly at odds with Frege's well-known platonism about numbers (as sets of equinumerous sets). It makes sense for other mathematical concepts. |
16878 | We must be clear about every premise and every law used in a proof [Frege] |
Full Idea: It is so important, if we are to have a clear insight into what is going on, for us to be able to recognise the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.212) | |
A reaction: Teachers of logic like natural deduction, because it reduces everything to a few clear laws, which can be stated at each step. |
16867 | Logic not only proves things, but also reveals logical relations between them [Frege] |
Full Idea: A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. Hence we find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.204) | |
A reaction: This is a key idea in Frege's philosophy, and a reason why he is the founder of modern analytic philosophy, with logic placed at the centre of the subject. I take the value of proofs to be raising questions, more than giving answers. |
16863 | Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege] |
Full Idea: Are there perhaps modes of inference peculiar to mathematics which …do not belong to logic? Here one may point to inference by mathematical induction from n to n+1. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
A reaction: He replies that it looks as if induction can be reduced to general laws, and those can be reduced to logic. |
16862 | The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege] |
Full Idea: Mathematics has closer ties with logic than does almost any other discipline; for almost the entire activity of the mathematician consists in drawing inferences. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
A reaction: The interesting question is who is in charge - the mathematician or the logician? |
16865 | 'Theorems' are both proved, and used in proofs [Frege] |
Full Idea: Usually a truth is only called a 'theorem' when it has not merely been obtained by inference, but is used in turn as a premise for a number of inferences in the science. ….Proofs use non-theorems, which only occur in that proof. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.204) |
16866 | Tracing inference backwards closes in on a small set of axioms and postulates [Frege] |
Full Idea: We can trace the chains of inference backwards, …and the circle of theorems closes in more and more. ..We must eventually come to an end by arriving at truths can cannot be inferred, …which are the axioms and postulates. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.204) | |
A reaction: The rival (more modern) view is that that all theorems are equal in status, and axioms are selected for convenience. |
16868 | The essence of mathematics is the kernel of primitive truths on which it rests [Frege] |
Full Idea: Science must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in this kernel. The essence of mathematics has to be defined by this kernel of truths. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.204-5) | |
A reaction: [compressed] I will make use of this thought, by arguing that mathematics may be 'explained' by this kernel. |
16871 | A truth can be an axiom in one system and not in another [Frege] |
Full Idea: It is possible for a truth to be an axiom in one system and not in another. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.205) | |
A reaction: Frege aspired to one huge single system, so this is a begrudging concession, one which modern thinkers would probably take for granted. |
16870 | Axioms are truths which cannot be doubted, and for which no proof is needed [Frege] |
Full Idea: The axioms are theorems, but truths for which no proof can be given in our system, and no proof is needed. It follows from this that there are no false axioms, and we cannot accept a thought as an axiom if we are in doubt about its truth. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.205) | |
A reaction: He struggles to be as objective as possible, but has to concede that whether we can 'doubt' the axiom is one of the criteria. |
16869 | To create order in mathematics we need a full system, guided by patterns of inference [Frege] |
Full Idea: We cannot long remain content with the present fragmentation [of mathematics]. Order can be created only by a system. But to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.205) |
16864 | If principles are provable, they are theorems; if not, they are axioms [Frege] |
Full Idea: If the law [of induction] can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
A reaction: This links Frege with the traditional Euclidean view of axioms. The question, then, is how do we know them, given that we can't prove them. |
9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege] |
Full Idea: Of any concept, we must require that it have a sharp boundary. Of any object it must hold either that it falls under the concept or it does not. We may not allow a third case in which it is somehow indeterminate whether an object falls under a concept. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.229), quoted by Ian Rumfitt - The Logic of Boundaryless Concepts p.1 n1 | |
A reaction: This is the voice of the classical logician, which has echoed by Russell. I'm with them, I think, in the sense that logic can only work with precise concepts. The jury is still out. Maybe we can 'precisify', without achieving total precision. |
6871 | We can't only believe things if we are currently conscious of their justification - there are too many [Goldman] |
Full Idea: Strong internalism says only current conscious states can justify beliefs, but this has the problem of Stored Beliefs, that most of our beliefs are stored in memory, and one's conscious state includes nothing that justifies them. | |
From: Alvin I. Goldman (Internalism Exposed [1999], §2) | |
A reaction: This point seems obviously correct, but one could still have a 'fairly strong' version, which required that you could always call into consciousness the justificiation for any belief that you happened to remember. |
6872 | Internalism must cover Forgotten Evidence, which is no longer retrievable from memory [Goldman] |
Full Idea: Even weak internalism has the problem of Forgotten Evidence; the agent once had adequate evidence that she subsequently forgot; at the time of epistemic appraisal, she no longer has adequate evidence that is retrievable from memory. | |
From: Alvin I. Goldman (Internalism Exposed [1999], §3) | |
A reaction: This is certainly a basic problem for any account of justification. It will rule out any strict requirement that there be actual mental states available to support a belief. Internalism may be pushed to include non-conscious parts of the mind. |
6874 | Internal justification needs both mental stability and time to compute coherence [Goldman] |
Full Idea: The problem for internalists of Doxastic Decision Interval says internal justification must avoid mental change to preserve the justification status, but must also allow enough time to compute the formal relations between beliefs. | |
From: Alvin I. Goldman (Internalism Exposed [1999], §4) | |
A reaction: The word 'compute' implies a rather odd model of assessing coherence, which seems instantaneous for most of us where everyday beliefs are concerned. In real mental life this does not strike me as a problem. |
6873 | Coherent justification seems to require retrieving all our beliefs simultaneously [Goldman] |
Full Idea: The problem of Concurrent Retrieval is a problem for internalism, notably coherentism, because an agent could ascertain coherence of her entire corpus only by concurrently retrieving all of her stored beliefs. | |
From: Alvin I. Goldman (Internalism Exposed [1999], §3) | |
A reaction: Sounds neat, but not very convincing. Goldman is relying on scepticism about short-term memory, but all belief and knowledge will collapse if we go down that road. We couldn't do simple arithmetic if Goldman's point were right. |
6875 | Reliability involves truth, and truth is external [Goldman] |
Full Idea: Reliability involves truth, and truth (on the usual assumption) is external. | |
From: Alvin I. Goldman (Internalism Exposed [1999], §6) | |
A reaction: As an argument for externalism this seems bogus. I am not sure that truth is either 'internal' or 'external'. How could the truth of 3+2=5 be external? Facts are mostly external, but I take truth to be a relation between internal and external. |
16876 | We need definitions to cram retrievable sense into a signed receptacle [Frege] |
Full Idea: If we need such signs, we also need definitions so that we can cram this sense into the receptacle and also take it out again. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.209) | |
A reaction: Has anyone noticed that Frege is the originator of the idea of the mental file? Has anyone noticed the role that definition plays in his account? |
16875 | We use signs to mark receptacles for complex senses [Frege] |
Full Idea: We often need to use a sign with which we associate a very complex sense. Such a sign seems a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we need what it contains. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.209) | |
A reaction: This exactly the concept of a mental file, which I enthusiastically endorse. Frege even talks of 'opening the receptacle'. For Frege a definition (which he has been discussing) is the assigment of a label (the 'definiendum') to the file (the 'definiens'). |
16879 | A sign won't gain sense just from being used in sentences with familiar components [Frege] |
Full Idea: No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.213) | |
A reaction: Music to my ears. I've never grasped how meaning could be grasped entirely through use. |
16873 | Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege] |
Full Idea: A thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.206) | |
A reaction: When such thoughts are treated as if the have objective (platonic) existence, I become bewildered. I take a thought (or proposition) to be entirely psychological, but that doesn't stop two people from having the same thought. |
16872 | A thought is the sense expressed by a sentence, and is what we prove [Frege] |
Full Idea: The sentence is of value to us because of the sense that we grasp in it, which is recognisably the same in a translation. I call this sense the thought. What we prove is not a sentence, but a thought. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.206) | |
A reaction: The 'sense' is presumably the German 'sinn', and a 'thought' in Frege is what we normally call a 'proposition'. So the sense of a sentence is a proposition, and logic proves propositions. I'm happy with that. |
16874 | The parts of a thought map onto the parts of a sentence [Frege] |
Full Idea: A sentence is generally a complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of a thought correspond to parts of the sentence. | |
From: Gottlob Frege (Logic in Mathematics [1914], p.207) | |
A reaction: This is the compositional view of propositions, as opposed to the holistic view. |
17366 | Virtually all modern views of speciation rest on relational rather than intrinsic features [Okasha] |
Full Idea: On all modern species concepts (except the phenetic), the property in virtue of which a particular organism belongs to one species rather than another is a relational rather than an intrinsic property of the organism. | |
From: Samir Okasha (Darwinian Metaphysics: Species and Essentialism [2002], p.201), quoted by Michael Devitt - Resurrecting Biological Essentialism 4 | |
A reaction: I am in sympathy with Devitt's attack on this view, for the same reason that I take relational explanations of almost anything (such as the mind) to be inadequate. We need to know the intrinsic features that enable the relations. |