11 ideas
6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
Full Idea: Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1) | |
A reaction: This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results. |
17744 | De Morgan started the study of relations and their properties [De Morgan, by Walicki] |
Full Idea: De Morgan started the sustained interest in the study of relations and their properties. | |
From: report of Augustus De Morgan (On the Syllogism IV [1859]) by Michal Walicki - Introduction to Mathematical Logic History D.1.1 |
13501 | De Morgan found inferences involving relations, which eluded Aristotle's syllogistic [De Morgan, by Hart,WD] |
Full Idea: There was a prejudice against relations (in favour of properties) but De Morgan and others that impeccable inferences turn on relations and elude Aristotle's syllogistic. Thus: All horses are animals. Hence, all heads of horses are heads of animals. | |
From: report of Augustus De Morgan (On the Syllogism IV [1859]) by William D. Hart - The Evolution of Logic 4 | |
A reaction: This is actually an early example of modern analytic philosophy in action. You start with the inferences, and then work back to the ontology and the definition of concepts. But in pinning down such concepts, do we miss their full meaning? |
6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |
Full Idea: Mathematical realism is the doctrine that mathematical objects exist, that much contemporary mathematics is true, and that the existence and truth in question is independent of our constructions, beliefs and proofs. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.12.9) | |
A reaction: As thus defined, I would call myself a mathematical realist, but everyone must hesitate a little at the word 'exist' and ask, how does it exist? What is it 'made of'? To say that it exists in the way that patterns exist strikes me as very helpful. |
6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
Full Idea: In maths the primary subject-matter is not mathematical objects but structures in which they are arranged; our constants and quantifiers denote atoms, structureless points, or positions in structures; they have no identity outside a structure or pattern. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.1) | |
A reaction: This seems to me a very promising idea for the understanding of mathematics. All mathematicians acknowledge that the recognition of patterns is basic to the subject. Even animals recognise patterns. It is natural to invent a language of patterns. |
6303 | Sets are positions in patterns [Resnik] |
Full Idea: On my view, sets are positions in certain patterns. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.5) | |
A reaction: I have always found the ontology of a 'set' puzzling, because they seem to depend on prior reasons why something is a member of a given set, which cannot always be random. It is hard to explain sets without mentioning properties. |
6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology. |
6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject. |
6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3) | |
A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component? |
6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points. | |
From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4) | |
A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns. |
1743 | The greatest deterrence for injustice is if uninjured parties feel as much indignation as those who are injured [Solon, by Diog. Laertius] |
Full Idea: Men can be most effectively deterred from committing injustice if those who are not injured feel as much indignation as those who are. | |
From: report of Solon (reports [c.600 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.So.10 |