Worth a look. — Banno
One can discover a truth without being the first to discover it (in this context); it is enough that one comes to believe it in an independent, reliable and rational way.
The difference between merely discovering a truth and proving it is a matter of transparency: for proving or following a proof the subject must be aware of the way in which the conclusion is reached and the soundness of that way; this is not required for discovery.
While there is no reason to think that mental arithmetic (mental calculation in the integers and rational numbers) typically involves much visual thinking,
form a number line representation [only] once we have acquired a written numeral system.
The Epistemology of Visual Thinking in Mathematics — Banno
As far as I am concerned, the only thing that really helps, are good examples, which are almost always lacking — alcontali
The Epistemology of Visual Thinking in Mathematics
. — Banno
And if someone does not see it thus, but sees it so...
...then it's not a justification at all. — Banno
for proving or following a proof the subject must be aware of the way in which the conclusion is reached and the soundness of that way; — Giaquinto
Does this say or show? — Banno
Does it differ significantly from fig 1?
I can't see how. — Banno
saying is a complicated way of showing. — Banno
Schematically put, in reasoning about things of kind K, once we have shown that from certain premisses it follows that such-and-such a condition is true of arbitrary instance c, we can validly infer from those same premisses that that condition is true of all Ks, with the proviso that neither the condition nor any premiss mentions c
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