It is said that mathematical axioms are neither true or false, but that they infers/imply their conclusions, and this is said by those with formalist leanings .


Will you not agree that if this is true, then it means that mathematics is based on unsound arguments if the axioms so used are false. And rarely do mathematical physicist check to see whether their axioms are true, rather they device a theory that fits(predicts) experiments(think Bohr model of the atom). It seem science uses this kind of reasoning a lot. There are no point particles nor isolated systems as envisioned by Newtons laws, yet these are foundations of classical mechanics. So there seem to be something false used to deduce something true. It seems unsound reasoning is pervasive in science, and I ask why should it be an issue especially in religious arguments? Why can’t we construct religious arguments using some hypothesis regardless of its truth.
–Johnson Mafoko

This is a much more interesting and complex question than your first. The basic concept is something I’ve studied for years, but unfortunately it is less straightforward than it seems at first.

You’re actually talking about two separate things, axiomatic systems and modeling.

In terms of the first, axioms are the starting point for any formal system. They are not considered “true” or “false” because truth is defined in relationship to the axioms.

When you say something is “true” in a formal system, what you really mean is that it is at least as guaranteed as the axioms.

In the case of mathematical systems such as Euclidian geometry, there is no necessary claim that the objects of the system have any physical reality, they exist only within the conceptual space defined by the axioms.

Physics, however, adds an additional claim that a given mathematical system “models” something real in the physical universe. In other words, if the mathematical system describes a certain numerical relationship between a triangle and a square, the claim is that a physical object in the shape of a triangle would have the same relationship with a physical object in the shape of a square.

It’s never a perfect match, and there is no way to definitively prove that the physical system follows all the same rules as it’s mathematical sibling. All that can be done is show by experiment that the measurements predicted by the mathematical system mirror the actual results produced by testing the physical system.

The advantage to all of this is that understanding the mathematical system allows you to make useful predictions about the physical system. For instance, the development of calculus allowed people to aim cannons with much greater accuracy. It was the practical results that convinced people of calculus’ worth, not because they had a metaphysical belief in infinitesimals. People tend to speak of scientific discoveries as if they were solid eternal truths, but what they really are is sets of tested, reliable, useful predictions.

Let me return to your query of why you can’t start from an unsure axiom and build towards a true conclusion when constructing a religious argument. You can certainly build a mathematics-like axiomatic theory of religion –many philosophers have done so. But none have had much success in showing that their system is a testable, reliable analog of the observable world.

For instance, in the Hindu religion, they believe in a system of karma where good deeds and bad deeds lead to rebirth in a better or worse existence. However, there is no good way to test the theory of karma, since one cannot –at least to my knowledge –reliably establish whether a given person is the reborn version of another one.

I myself am deeply interested in creating a testable, reliable theory of morality that aligns with my religious and metaphysical commitments. But it is far from being an easy task.

Just recently saw the following argument in a logic book: all lions are herbivores all zebras are lions ————– therefore all zebras are herbivores this seems to be logically valid syllogism, but it is disturbing.

I have been reading your site, and there is somewhere you said a conclusion can logical valid but unsound. Is the following argument valid but unsound? I am not sure about what unsound arguments mean? Can you please clarify this for me. – Johnson Mafoko

Yes, that is a valid, unsound argument. The structure is good, but the content is bad. This is the case even though the conclusion is correct.

The way it works is this:

Invalid means the structure is bad. There are no benefits to an invalid argument, the premises have no meaningful connection to the conclusion.

Valid means the structure is good. If an argument is valid, it means the conclusion is at least as good as the premises. So if you put in true premises, you get a true conclusion. However, it doesn’t mean that if you put in false premises you get a false conclusion. In logic, false premises can lead to any conclusion, even when the argument structure is valid.

Sound means that the argument is valid and that the premises are true. A sound argument will guarantee a true conclusion. It is the only type of argument that guarantees a true conclusion.

Please note that only “formal” arguments –the kind of very artificial, highly structured arguments found in logic books and dealing only with unambiguously true or false statements –can be either valid or sound. (Different terms are used for less formal arguments).