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Sections 5 and 7 will describe a few possible reasons for a fallibilist to regard your belief that 2 + 2 = 4 as being fallible. In the meantime, we need only note schematically how F* would accommodate those possible reasons. The basic approach would be as follows. Although your belief that 2 + 2 = 4 cannot be false (once it is present), your supposed justification for it is fallible. This could be so in a few ways. For a start, maybe you are merely repeating by rote something you were told many years ago by a somewhat unreliable school teacher. (Imagine the teacher having been poor at making accurate claims within most other areas of mathematics. Even with respect to the elements of mathematics about which she was accurate, she might have been merely repeating by rote what she had been told by her own early — and similarly unreliable — teachers.) The fallibility of memory is also relevant: over the years, one forgets much. Still, your current belief that 2 + 2 = 4 seems accurate. And it need not be present only because of your fallible memory of what your fallible teacher told you. Suppose that you are now very sophisticated in your mathematical thinking: in particular, your justification for your belief that 2 + 2 = 4 is purely mathematical in content. That justification involves clever representation, via precisely defined symbols, of abstract ideas. Nevertheless, even such purely mathematical reasoning can mislead you (no matter that it has not done so on this occasion). Really proving that 2 + 2 = 4 is quite difficult; and when people are seeking to grasp and to implement such proofs, human fallibility may readily intrude. Actual attempts to establish mathematical truths need not always lead to accurate or true beliefs.

Remove thesis attribution link from the footer

remove thesis attribution link from the footer

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