Well, existentially, a “Number” CAN only refer to a “name”. That name can refer to the base number set (in base ten, that’s 0-9), or it can refer to a positional number ( say, 12,345 ) or it can refer to a FUNCTION (say .9, 1/2, or the square root of 2, or any other non-reducible function.)
So the only ‘numbers’ that can exist are the natural numbers. The rest of the so called ‘numbers’ must refer to functions (a sequence of operations).
We can however, speak in terms of ‘meaning’ not ‘truth’. This is what mathematicians do. They speak in meaning. But we can translate ‘meaning’ into ‘truth’ (parsimonious existential necessity) for all mathematical statements. And as such, we can restate mathematics in truthful, existential, natural language.
But this would be burdensome. Because the reason that mathematicians use ‘meaning’ is to make the best use of symbols to save both state, and the possible operations upon that state. In other words, just as we use analogies (meaning) to simplify our effort at communication, mathematicians use symbols to simplify the difficulty in saving and transforming ratios.
Now, once we embark upon pragmatism (analogy, meaning) and not names (names) we get into the problem error, conflation, bias, wishful thinking, suggestion, and deceit.
But as long as we can still translate meaning into truthfulness we can test our statements (series of symbols) against error conflation, bias, wishful thinking, suggestion and deceit. But then if we communicate by meaning, one either takes responsibility for the recipient’s interpretation, or one is exporting the cost of testing our statements to the other parties.
In other words, if we take a discount on communication, we force the cost of falsification upon others. Which may be beneficial for both sides. On the other hand, if one commits error, conflation, bias, wishful thinking, suggestion or deceit, then how do we know whether which of those that one engaged in?
Especially when many thinkers have seeded humanity with vast costly falsehoods by speaking meaningfully but not truthfully.
Source date (UTC): 2016-05-04 14:04:00 UTC