Tacit Assumptions, Tacit Knowledge
An alert reader may have noticed that writing symbols □, ■□, ■■□, ■■■□, etc. by concatenating a marked square, i.e. ■, to an existing string is different than writing the completed symbols one after another on a Turing-machine tape. Another entirely possible scenario would be to generate the symbol-strings one after another on different sections of tape e.g. after three symbols: ■■■□■■□■□□. The proof that these two possibilities are different is easy: they require different "programs". But in a sense both versions create the same objects; in the second case the objects are preserved on the tape. In the same way, if a person were to write 0, then erase it, write 1 in the same place, then erase it, write 2, erase it, ad infinitum, the person is generating the same objects as if they were writing down 0 1 2 3 ... writing one symbol after another to the right on the paper.
Once the step has been taken to write down the symbols 3 2 1 0 one after another on a piece of paper (writing the new symbol on the left this time), or writing ∫∫∫※∫∫※∫※※ in a similar manner, then putting them in 1-1 correspondence with the Turing-tape symbols seems obvious. Digging holes one after the other, starting with a hole at "the origin", then a hole to its left with one pebble in it, then a hole to its left with two pebbles in it, ad infinitum, raises practical questions, but in the abstract it too can be seen to be conducive to the same 1-1 correspondence.
However, nothing in particular in the definition of genetic versus axiomatic methods clears this up—these are issues to be discussed in the metatheory. The mathematician or scientist is to be held responsible for sloppy specifications. Breger cautions that axiomatic methods are susceptible to tacit knowledge, in particular, the sort that involves "know-how of a human being" (Breger 2000:227).
Read more about this topic: Object Theory
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