This is another brief, but important, note within our discussion of evaluating claims that people make while trying to prove a point. The shortest way to say this is that there is more than one type of claim. You should evaluate claims based upon the category into which they fit. I could see this summary statement being a little too vague unless you can see what I mean, so let me explain.
As a mathematician, I am trained to look at mathematics with certain standards. I am taught to rely on nothing more than definitions, foundational mathematical axioms, and the basic rules of logic. Because of the goal of mathematicians – to know with logical rigor things that are true about the mathematical realm – this is an entirely appropriate way to look at mathematics. I say this is appropriate because this way of looking at mathematics is designed specifically to achieve the goal of knowing with logical rigor the correct answer. In fewer, words, the method fits the goal.
Why do I point this out? Well, not every field of academia uses the same standard. For instance, if I were to use my mathematical standards while studying history, I would learn literally nothing – I would just point out that it is at least possible that someone made everything up, and therefore on the grounds of the rules of logic I would reject all of history – even events that happened five minutes ago! If I apply mathematical standards to studies in literature, then the exact same thing will happen. Even if I apply mathematical standards to science, I will accuse the scientist of fallacy, since empirical evidence has no place within the rules of logic. In fact, but using these standards I will destroy all of human knowledge except for mathematics itself and a couple areas of philosophy.
This should not tell you that everything outside of mathematics ought to be placed in doubt. It just tells you that the standard used by mathematicians is not appropriate outside of mathematics. There are plenty of other standards outside of mathematics that are appropriate in different contexts. In a courtroom, the standard used for evidence is that it point to its conclusion beyond a reasonable doubt. This is still a very high bar, but it isn’t as high as the mathematical standard. Most scientific disciplines have the standards of empirical verification of theoretical predictions – which is significantly different than either the courtroom of mathematics. When studying history, you use various kinds of historical techniques that are entirely inappropriate anywhere outside of historical studies.
So, if we are evaluating a claim that someone else makes, we must be very careful to understand what kind of claim they are making, and what kinds of evidence are relevant to the question. It is possible that an idea will have multiple dimensions – for example, evaluating ancient history involves, geography, literary studies, anthropology, historical studies, and archaeology (at least these – perhaps even more). Each of these areas has their own methods and techniques that are designed to be helpful at answering very specific kinds of questions, and we should always be careful that we are taking all of this into account.
Of course, in order to do this we must take some time to learn about various kinds of techniques and learning about where those techniques are viewed as relevant by academic professionals. If you want to learn about whether a certain mathematical technique is valid, you ask a mathematician, not a scientist, because there are statements known by mathematicians to be false that the scientific method would end up regarding as true, because counterexamples to these statements, while we know they exist, are beyond the power of modern computers to locate. A scientific approach will get you a lot of the right ideas, but a lot of wrong ones too. If you want to know about scientific techniques, mathematicians and historical studies can be helpful, but you’re going to learn the most from a scientist. The same goes for everything you can think of that humans study today.
This is why I take so much time attempting to explain careful methods of thinking. We will often go astray if we use the scientific method when it isn’t appropriate, or if we apply a non-scientific methodology in science. We must be careful if we want to know the truth about reality, because the deeper truths about reality are not always simple.