From this general recommendation, we draw out two design principles for promoting mathematical proof: (1) eliciting dynamic depictive gestures through relevant directed actions that are congruent with the geometric relations that are present in the conjecture tasks; and (2) activating language systems that are explicitly coordinated with the relevant action systems, through pedagogical cues.
In the sense that the sheer postulation of certain principles is enough for mathematical practice: 'A mathematical subject with its accompanying posits can be created ex nihilo by simply writing down a set of axioms' (Azzouni 2004, p. 127).
And this is what allows the neo-Fregean logicist to derive the principle of mathematical induction for the natural numbers.
Hence result is true for N. Thus by principle of mathematical induction the result is true for any natural number N. For any m > 0, replacing f i by f m + i and x i by x m + i in Lemma 3.1 we get the following result: Let ( X, d ) be a compact metric space and F = { f n } n = 0 ∞ be a time varying map on X, where family { f n } is equicontinuous on X and N is a natural number.
end{aligned} Thus (3.2) is true for (n=r+1), and hence by the principle of mathematical induction it is true for all n.
Our mathematical intuition provides intrinsic evidence for mathematical principles.
Aside from intrinsic evidence, it is in Gödel's view also possible to obtain extrinsic evidence for mathematical principles.
Therefore, by the principle of mathematical induction, (2.6) holds true for all n ∈ N. □.
Hence, by the principle of mathematical induction, (2.1) is true for all n ∈ N. □.
In indirect contexts sense, and not designation, matters and so we may know the well-ordering principle for natural numbers, but not know the principle of mathematical induction because, while they are equivalent in truth value, they have different senses.
