I think the word "based" has become debased. Too often does one find "X is based on Y", particularly, "X is based on the principle of Y", though the misuse of the word "principle" deserves an article its own.
"based" could mean all sorts of things, which I shall illustrate by examples taken from the first few pages of Google hits for "is based on". Derivation: "Ubuntu is based on Debian"; Logical dependency: "critique of ethanol is based on flawed findings"; many things in between, and a sort of relationship I find it hard to characterise: "Newborn's face recognition is based on low spatial frequencies".
It's the "many things in between" which it's hard to deal with. For example: "Islam's horror of homosexuality is based on Koranic misinterpretation". "Modern `Commercial Law' is based on Ancient Babylonian Codes". With these two assertions, some sort of relationship is asserted between a contemporary phenomenon and a rule written down centuries ago and authoritative for some community in the Middle East. Is it that the old rule entails the new phenomenon, or that the new phenomenon somehow incorporates it in part? And if we get to the bottom of that, what do we make of facial recognition's relation to frequencies? Or "America's trade is based on protectionism", "Restorative justice is based on community involvement", "Atheism is based on faith", "Anti-Stalinism is based on Nazi Lies", "Quantitative easing is based on discredited economics"? In particular, if the thing which is the basis turns out to be false, or gets destroyed, etc, what is the implication for something which is "based" thereon?
Obviously, no sort of logical dependency is intended by the author of "trade is based on protectionism", after all trade is an activity and protectionism is a set of normative propositions. If protectionism is taken to refer to the practice of protectionist policies, then what would it mean for trade to be "based" thereon? Does it mean "benefit from in part", "necessarily require", "possess as historical antecedent"? And which of these sense of "based on" accounts for those other four examples? In the case of quantitative easing, what are we to make of it? A policy might work in practice, even if undertaken by people motivated by some incorrect theory, yet this possibility is easy to discount, as by the use of two simple words one may contrive to say, "X is logically dependent on, or historically derived from, or necessarily requires Y", which entails that any practice whose success cannot currently be explained may be discredited by a conflation of its historical or logical antecedents.
Scientific and mathematical reasoning are much more different from each other than people intuitively realise: they differ in what sort of ideas they start with, what consequence erroneous reasoning has, and the structure of the network of ideas they create.
Science starts with a very small set of assumptions, basically that the whole of the observable universe behaves in a regular, mechanical fashion. New scientific knowledge is created when we make a set of observations and infer a general rule to which these observations conform. The inputs are mostly facts about the observable world around us, and existing scientific theories. The direction of inference is inductive, from the specific to the general. This means that if enough of the specific observations are for some reason wrong, then the generalisation inferred from them could be wrong too.
That is not how maths works: you start with a set of assumptions and without any observations. Indeed: there is no observation you can make of the physical universe which can help you prove or disprove a mathematical theory. Maths starts and ends with objects and concepts which are not physical things we can observe. There is, however, a much more important distinction: mathematical reasoning generally proceeds from the general to the specific, and is like a chain, rather than a rope: if a single assumption or inference is wrong, everything which depends on it is probably wrong too.
People who let their political views get in the way of their ability to think often exult when a theory they dislike is disproven. We don't know what level of academic fraud and incompetence exists, and occasionally the mainstream media covers mistaken results in, e.g., climatology or economics. These disciplines, whatever their status within the sciences are sciences, not branches of maths, yet highly intelligent people seem to believe that some disproven major claim in one of these disciplines invalidates large swathes of results elsewhere in them. That's not how non-mathematical knowledge is structured: it's a rope with many plies, not a chain as weak as its weakest link.
(I'd include theological reasoning alongside maths, as starting from a set of assumptions (e.g., the contents of the Bible), and legal reasoning, at least in the common law and Sharia, as being more like science)