So within our model of mathematics we might say that things are “true” if they can be derived, and are “false” if they lead to an “explosion”. In other words, introducing a single “false axiom” will start an explosion that will eventually “blow up everything”. There is another aspect to this, discussed since at least the Middle Ages, often under the name of the “ principle of explosion”: that as soon as one assumes any statement that is false, one can logically derive absolutely any statement at all. So, for example, the single statement 1+1=2 is true, but the infinite collection of statements 1+1= n for any other n are all false. And perhaps this is not surprising -because in a sense if there’s one true statement about something there are typically an infinite number of false statements about it. But one feature of our view of mathematics here is that actually truth and falsity seem to have a rather different character.
In traditional logic it has tended to be assumed that truth and falsity are very much “the same kind of thing” -like 1 and 0. But to get further in discussing things like this we really need not only to have a notion of truth, but also a notion of falsity. One issue that can come up is that our axioms might be inconsistent -in the sense that from them we can derive two obviously inconsistent statements. And if we’re starting from axioms, then in a sense we’ll never have an “absolute notion of truth” -because whatever we derive is only “as true as the axioms we started from”. In other words, we’ve been more concerned with “structurally deriving” that “ 1+1=2” than in saying that “ 1+1=2 is true”.īut what is the relation between this kind of “constructive derivation” and the logical notion of truth? We might just say that “if we can construct a statement then we should consider it true”. But the way we’ve modeled mathematics here has been much more about what statements can be derived (or entailed) than about any kind of abstract notion of what statements can be “tagged as true”. And when logic was applied to the foundations of mathematics, “truth” was also usually assumed to be quite central.
#FALSE NOTION DEFINITION HOW TO#
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