I argued, elsewhere in this thread, that you could not evolve a mathematical understanding that 2+2=3, because “2+2=4” is mandatory, independent of any physical reality or observation, any context whatsoever. This was not whole-heartedly accepted so let me again try to defend the inviolability and universality of mathematics.
I assert my case with three definitions: a definition of mathematics, a definition of logically follows, and a definition of consistent.
Mathematics is: you define things and then determine what logically follows from those things.
As fallible humans, we may not know if something really logically follows. But we just define what we mean by “logically follow”: logically following means follows necessarily, mandatorily, independently of everything else.
If an axiomatic system is consistent it means that if x logically follows, then “not x” does not logically follow.
Maybe you don’t like these definitions. But that is what they are. To the extent to which I can speak for mathematics, if you change these definitions, you’re not talking about mathematics any more.
Unlike science, the definitions precede the observation. Here, imagine: humans in the savannas, dressed in their animal skins, shake their clubs at the empirical world and the unreliability of the senses. They wield the power by defining exactly what they mean with no heed whatsoever to what actually is. This is the difference between math and science, this is why mathematics is trustworthy even when all else—all empirical sense—might fail.
Consider Peano arithmetic. We understand that 2+2=4 logically follows from the axioms in Peano arithmetic in our current universe, context C1. If “2+2 is not equal to 4” is deduced by the Peano axioms in some other context C2, then at least one of the following must be true:
(a) 2+2=4 does not logically follow in context C2, so 2+2=4 did not really logically follow from the Peano axioms—by definition of logically follow, as it is context independent,
(b) 2+2=4 does logically follow in context C2, so the Peano arithmetic is not consistent -- by definition of consistent
Thus, mathematical truth is independent of context, including the physical world. This is why mathematicians love saying, “it is true by definition”. It is the precise source of the omnipotence of mathematics. That’s it. Everything mathematically true is true by definition. It’s our terms, our game. Unlike the empirical world where we don’t get to define anything, where the existence of things precede our observation.
If an axiomatic system is consistent it means that if x logically follows, then “not x” does not logically follow.
While this might seem overly pedantic (hardly something to complain about in this discussion), I’d like to point out that this definition only matches the usual one if you also accept the law of non-contradiction. More precisely, a system is consistent if it does not contain a contradiction.
Also, your definitions of Mathematics and “logically follows” don’t seem very… good. Did you make them up?
EDIT: changed “excluded middle” to “non-contradiction”. duh.
More precisely, a system is consistent if it does not contain a contradiction.
What we mean by a contradiction is the truth of a statement of the form “x and not x”, or, in Bynerma’s terms, that there is a statement x such that both x and “not x” logically follow (from the axiomatic system). (Things that we (correctly) intuitively recognize as contradictions allow us to derive a statement of that form.) So, although Bynerma used slightly unusual language, she correctly expressed the content of the usual definition.
The Law of the Excluded Middle is not needed here. It says that statements of the form “x or not x” are true (provable even, according to Wikipedia).
I see what I was doing here. The law of excluded middle is equivalent to the law of non-contradiction. Probably why I had the two confused. Example in sentential logic:
I argued, elsewhere in this thread, that you could not evolve a mathematical understanding that 2+2=3, because “2+2=4” is mandatory, independent of any physical reality or observation, any context whatsoever. This was not whole-heartedly accepted so let me again try to defend the inviolability and universality of mathematics.
I assert my case with three definitions: a definition of mathematics, a definition of logically follows, and a definition of consistent.
Mathematics is: you define things and then determine what logically follows from those things.
As fallible humans, we may not know if something really logically follows. But we just define what we mean by “logically follow”: logically following means follows necessarily, mandatorily, independently of everything else.
If an axiomatic system is consistent it means that if x logically follows, then “not x” does not logically follow.
Maybe you don’t like these definitions. But that is what they are. To the extent to which I can speak for mathematics, if you change these definitions, you’re not talking about mathematics any more.
Unlike science, the definitions precede the observation. Here, imagine: humans in the savannas, dressed in their animal skins, shake their clubs at the empirical world and the unreliability of the senses. They wield the power by defining exactly what they mean with no heed whatsoever to what actually is. This is the difference between math and science, this is why mathematics is trustworthy even when all else—all empirical sense—might fail.
Consider Peano arithmetic. We understand that 2+2=4 logically follows from the axioms in Peano arithmetic in our current universe, context C1. If “2+2 is not equal to 4” is deduced by the Peano axioms in some other context C2, then at least one of the following must be true:
(a) 2+2=4 does not logically follow in context C2, so 2+2=4 did not really logically follow from the Peano axioms—by definition of logically follow, as it is context independent,
(b) 2+2=4 does logically follow in context C2, so the Peano arithmetic is not consistent -- by definition of consistent
Thus, mathematical truth is independent of context, including the physical world. This is why mathematicians love saying, “it is true by definition”. It is the precise source of the omnipotence of mathematics. That’s it. Everything mathematically true is true by definition. It’s our terms, our game. Unlike the empirical world where we don’t get to define anything, where the existence of things precede our observation.
While this might seem overly pedantic (hardly something to complain about in this discussion), I’d like to point out that this definition only matches the usual one if you also accept the law of non-contradiction. More precisely, a system is consistent if it does not contain a contradiction.
Also, your definitions of Mathematics and “logically follows” don’t seem very… good. Did you make them up?
EDIT: changed “excluded middle” to “non-contradiction”. duh.
ETA: Yeah, my point was stupid. Nevermind.
What we mean by a contradiction is the truth of a statement of the form “x and not x”, or, in Bynerma’s terms, that there is a statement x such that both x and “not x” logically follow (from the axiomatic system). (Things that we (correctly) intuitively recognize as contradictions allow us to derive a statement of that form.) So, although Bynerma used slightly unusual language, she correctly expressed the content of the usual definition.
The Law of the Excluded Middle is not needed here. It says that statements of the form “x or not x” are true (provable even, according to Wikipedia).
Absolutely correct—note edit
I see what I was doing here. The law of excluded middle is equivalent to the law of non-contradiction. Probably why I had the two confused. Example in sentential logic:
¬(p∧¬p) (law of non-contradiction)
¬p∨¬¬p | 1, de morgan’s
¬p∨p | 2, double negation
p∨¬p | 3, commutation (law of excluded middle)