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6 ideas
9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
Full Idea: Mathematics seems to be the one and only place where we humans can be absolutely sure that we got it right. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
A reaction: Apart from death and taxes, that is. Personally I am more certain of the keyboard I am typing on than I am of Pythagoras's Theorem, but the experts seem pretty confident about the number stuff. |
9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
Full Idea: 'There are two apples' can be recast as 'x is an apple and y is an apple, and x isn't y, and if z is an apple it is the same as x or y', which makes no appeal at all to mathematics. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: He cites this as the basis of Hartry Field's claim that science can be done without numbers. The logic is ∃x∃y∀z(Ax&Ay&(x¬=y)&(Az→z=x∨z=y)). |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
Full Idea: 'Analysis' is the theory of the real numbers. | |
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
15364 | English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten] |
Full Idea: The number of English expressions is denumerably infinite. But Cantor's theorem can be used to show that there are nondenumerably many real numbers. So not every real number has a (simple or complex name in English). | |
From: Leon Horsten (The Tarskian Turn [2011], 06.3) | |
A reaction: This really bothers me. Are we supposed to be committed to the existence of entities which are beyond our powers of naming? How precise must naming be? If I say 'pick a random real number', might that potentially name all of them? |
9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
Full Idea: The number π is not only irrational, but it is also (unlike √2) a 'transcendental' number, because it is not the solution of an algebraic equation. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
A reaction: So is that a superficial property, or a profound one? Answers on a post card. |
9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
Full Idea: Mathematics hooks onto the world by providing representations in the form of structurally similar models. | |
From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
A reaction: This is Brown's conclusion. It needs notions of mapping, one-to-one correspondence, and similarity. I like the idea of a 'model', as used in both logic and mathematics, and children's hobbies. The mind is a model-making machine. |