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7 ideas
9812 | In mathematics, if a problem can be formulated, it will eventually be solved [Badiou] |
Full Idea: Only in mathematics can one unequivocally maintain that if thought can formulate a problem, it can and will solve it, regardless of how long it takes. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.17) | |
A reaction: I hope this includes proving the Continuum Hypothesis, and Goldbach's Conjecture. It doesn't seem quite true, but it shows why philosophers of a rationalist persuasion are drawn to mathematics. |
9965 | There couldn't just be one number, such as 17 [Jubien] |
Full Idea: It makes no sense to suppose there might be just one natural number, say seventeen. | |
From: Michael Jubien (Ontology and Mathematical Truth [1977], p.113) | |
A reaction: Hm. Not convinced. If numbers are essentially patterns, we might only have the number 'twelve', because we had built our religion around anything which exhibited that form (in any of its various arrangements). Nice point, though. |
9813 | Mathematics shows that thinking is not confined to the finite [Badiou] |
Full Idea: Mathematics teaches us that there is no reason whatsoever to confne thinking within the ambit of finitude. | |
From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.19) | |
A reaction: This would perhaps make Cantor the greatest thinker who ever lived. It is an exhilarating idea, but we should ward the reader against romping of into unrestrained philosophical thought about infinities. You may be jumping without your Cantorian parachute. |
9966 | The subject-matter of (pure) mathematics is abstract structure [Jubien] |
Full Idea: The subject-matter of (pure) mathematics is abstract structure per se. | |
From: Michael Jubien (Ontology and Mathematical Truth [1977], p.115) | |
A reaction: This is the Structuralist idea beginning to take shape after Benacerraf's launching of it. Note that Jubien gets there by his rejection of platonism, whereas some structuralist have given a platonist interpretation of structure. |
9963 | If we all intuited mathematical objects, platonism would be agreed [Jubien] |
Full Idea: If the intuition of mathematical objects were general, there would be no real debate over platonism. | |
From: Michael Jubien (Ontology and Mathematical Truth [1977], p.111) | |
A reaction: It is particularly perplexing when Gödel says that his perception of them is just like sight or smell, since I have no such perception. How do you individuate very large numbers, or irrational numbers, apart from writing down numerals? |
9962 | How can pure abstract entities give models to serve as interpretations? [Jubien] |
Full Idea: I am unable to see how the mere existence of pure abstract entities enables us to concoct appropriate models to serve as interpretations. | |
From: Michael Jubien (Ontology and Mathematical Truth [1977], p.111) | |
A reaction: Nice question. It is always assumed that once we have platonic realm, that everything else follows. Even if we are able to grasp the objects, despite their causal inertness, we still have to discern innumerable relations between them. |
9964 | Since mathematical objects are essentially relational, they can't be picked out on their own [Jubien] |
Full Idea: The essential properties of mathematical entities seem to be relational, ...so we make no progress unless we can pick out some mathematical entities wihout presupposing other entities already picked out. | |
From: Michael Jubien (Ontology and Mathematical Truth [1977], p.112) | |
A reaction: [compressed] Jubien is a good critic of platonism. He has identified the problem with Frege's metaphor of a 'borehole', where we discover delightful new properties of numbers simply by reaching them. |