Believe me, no mathematician “declares axioms to be true”, they are just initial statements, Aristotle would call premises, from which you derive other statements using the rules of logic.
They are initial statements from which we derive other statements, yes. But they are known to be true not because of faith or evidence, but simply by fiat.
Mathematicians are divided in their language concerning this topic. Some of them will, as I have, freely acknowledge that axioms are true by declaration. For example, cf. Norman Bigg’s
, p25Discreet Mathematics, and Stephen Cole Kleene’s
, p25Mathematical Logic, who conform to this practice. Other mathematicians talk of axioms being “self-evidently true.” But of course I dislike this terminology. Self-evidently true in what respect? For example, consider the ruler axiom:
For any line
m, there is a function
f :
m o
R that is one-to-one and onto, and such that for all points
X,
Y on
m, we have
XY=|
f(
X)-
f(
Y)|.
There is no proof for this. We simply assume it to be true in plane geometry. Is it self-evidently true? I find that hard to swallow.
But regardless, you will find mathematicians who talk that way. But even they will admit that ultimately, axioms are true because we say they are, within a particular formal system.
Not the axioms themselves but the conclusions of the theory built on them might say something about the physical world (c.f. Eugene Wigner’s “unreasonable effectiveness of mathematics”).
The proved theorems in math say absolutely nothing
directly about the real world. We can only try to use them to make fallible predictions about the behavior of matter.
It is one thing to check the internal coherence of a mathematical model (of some physical) reality, and another to check its applicability. In the same sense, a world-view built on religious beliefs needs to be checked only insofar as it says something about observable physical reality. This however usually does not involve the initial “axioms” of a belief system, only its applications.
Indeed, Einstein’s famous aphorisma: “As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality” could be countered by: “As far as religious symbols (and norms) refer to observable reality (and rules that can be enforced) they are not certain; as far as they are certain they do not refer to observable reality (and rules that can be enforced).”
I agree that you have to follow whatever “empiricism demands” of you but please accept that other people subscribe to other philosophical systems.
Of course I accept that. I merely remind you that those “other philosophical systems” are often quite nutty.
