9830
|
Bolzano began the elimination of intuition, by proving something which seemed obvious [Bolzano, by Dummett]
|
|
Full Idea:
Bolzano began the process of eliminating intuition from analysis, by proving something apparently obvious (that as continuous function must be zero at some point). Proof reveals on what a theorem rests, and that it is not intuition.
|
|
From:
report of Bernard Bolzano (Theory of Science (Wissenschaftslehre, 4 vols) [1837]) by Michael Dummett - Frege philosophy of mathematics Ch.6
|
|
A reaction:
Kant was the target of Bolzano's attack. Two responses might be to say that many other basic ideas are intuited but impossible to prove, or to say that proof itself depends on intuition, if you dig deep enough.
|
6424
|
Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
|
|
Full Idea:
The Formalists, led by Hilbert, maintain that arithmetic symbols are merely marks on paper, devoid of meaning, and that arithmetic consists of certain arbitrary rules, like the rules of chess, by which these marks can be manipulated.
|
|
From:
Bertrand Russell (My Philosophical Development [1959], Ch.10)
|
|
A reaction:
I just don't believe that maths is arbitrary, and this view pushes me into the arms of the empiricists, who say maths is far more likely to arise from experience than from arbitrary convention. The key to maths is patterns.
|
6425
|
Formalism can't apply numbers to reality, so it is an evasion [Russell]
|
|
Full Idea:
Formalism is perfectly adequate for doing sums, but not for the application of number, such as the simple statement 'there are three men in this room', so it must be regarded as an unsatisfactory evasion.
|
|
From:
Bertrand Russell (My Philosophical Development [1959], Ch.10)
|
|
A reaction:
This seems to me a powerful and simple objection. The foundation of arithmetic is that there are three men in the room, not that one plus two is three. Three men and three ties make a pattern, which we call 'three'.
|