Re: Only humans have souls? (Score 1)

Let's see. Where to start?

Axioms are not accepted on faith in any sense of the word. They are taken as given to give a framework in which one can make mathematical deductions. Deductive logic can take you from one idea to another and show that one idea logically entails another. Where do we start? From the axioms.

The results we can derive depend on what axioms we accept. What axioms we accept differ from instance to instance as we try to solve different classes of problems. In Euclidean geometry, we accept as an axiom that given a line and a point not on that line, there is only one line through that point which is parallel to the original line. Any results that we derive based on that axiom are then only necessarily true when that axiom holds. Thus, if I were designing a computer game that took place in a Euclidean world, I could apply those results to my heart's content because the axiom holds in that instance.

On the other hand, if I were to try solving problems in General Relativity, where that axiom does not hold, I could not use the results which I had derived.

One can (and does) change the axioms one accepts as long as all of the axioms accepted in a particular instance are mutually consistent. The axioms you hold for doing real-number arithmetic are different from the ones you use when computing with (say) Clifford algebras.

For example, real numbers are commutative with respect to multiplication. a*b = b*a when both a and b are real numbers. The commutative property of real numbers is only an axiom that we accept because we can get useful results out of that axiom. Rather than an axiom's truth being defined by whether or not a*b actually does equal b*a, the real numbers are in part defined by that axiom.

You can generate a Clifford algebra by not accepting this axiom (and adding a few desiderata into the mix). Thus, in general a*b != b*a in a Clifford algebra. This property does not make Clifford algebras untrue. It is simply that we don't use a particular axiom.

The logical truth of a statement can only be deduced from the background, as it were. What is true against one background of axioms may not be against another.

Now, back to science. Science uses mathematics as a tool. Which axioms a scientists accepts for a particular calculation depends on what axioms he can realistically apply to the situation. When measuring the volume of a room, for instance, one accepts that the axioms that apply to real numbers apply to the length measurements one makes. The axioms may not actually hold because, for example, space may be quantized and not continuous, but our errors in physical measurement make such a distinction pointless. To the limit of the accuracy of our physical measurements, length quantities seem to obey the axioms of real numbers. We can then apply our commutativity axiom to show that V = l*w*h = w*h*l = ...

Should evidence show that an axiom does not hold in a particular instance, that axiom its consequences are dropped and work continues with the axioms that do hold. For example, we had once believed that space obeyed the axioms of Euclidean geometry. Early this century, we found that we were wrong, and we had to replace the non-applicable axioms with others.

I think I'll stop now. This is off-topic.