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)
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