In our time there is a tendency to treat Mathematics as though it is a natural science.
This reflects in part the shift in meaning of the term Science in recent centuries, from knowledge and systematic bodies of more or less established knowledge, to the natural sciences based on inductive reasoning on observation and experiment. Where, inductive here denotes arguments whereby evidence — typically empirical — supports but does not logically demonstrate a conclusion, as a rule provisionally. Such has been multiplied by Scientism, the view, assumption or implication that Science ring fences and monopolises reliable, serious knowledge. (Of course, such Scientism is self-referentially incoherent as this is an epistemological and thus philosophical claim; it fails its own test.)
In a current thread, this has led to an exchange, worth headlining for reflection:
>>KF, 4: . . . PS: I have also recently seen someone describing Mathematics as a “Science” — a sign of Scientism’s attempt to monopolise all serious knowledge. Of course Scientism is itself self-referentially incoherent. However, more importantly, Mathematics is precisely not a discipline in which theoretical constructs are empirically tested and are taken as a sort of weak form, provisional knowledge due to empirical reliability. We need sterner stuff, rooted in logic and coherence, driven in the end by self-evident first principles of right reason. For example, number itself pivots on distinct identity, e.g. A vs ~A leads to 1 and 2 etc. Indeed, this pattern of being rooted in logic is part of why Mathematics plays the role of a plumbline in considerations on scientific endeavours. We need the logic of structure and quantity (including space etc) to be a standard of reference. That we can do Mathematics is a sign.
BO’H, 5: Eh? The four colour map theorem was a theoretical construct that was empirically tested: they narrow down the possible maps, and then used a computer to literally try every combination. Other postulates are certainly “a sort of weak form, provisional knowledge due to empirical reliability”, because not every combination can be tested, and no other proofs are available (e.g. Golbach’s conjecture).
KF, 6: . . . mathematics has an emphasis on axiomatic systems and on results derived therefrom by logical, step by step proofs, accumulating into what is now a huge body of knowledge. You know full well that proofs are generally not done by empirical examples and making an inference to generality or to the best current explanation or the like. [NB: I made an error regarding BO’H’s background, corrected later; not material.]
BO’H, 9: . . . I gave an example @5 where mathematics was done empirically. You just ignored that comment . . . [Actually, posting difficulties — Internet access here is spotty right now — and events intervened, then I thought I could wait.]
KF, 10: . . . proof of a finite result by complete enumeration is not the same as an empirical, inference to best explanation or generalisation from a consistent pattern, inductive argument. It is like proving a logical conclusion by truth table based examination of cases instead of doing the algebra of logic.
EMH, 11: The material universe is finite and discrete. Mathematics is infinite. Therefore, the material universe is not all that exists. Furthermore, anything that can do math cannot have a material origin. Thus, it is impossible to explain our ability to do math by evolution. This is one of those things that is so obvious materialists just ignore it.
JAD, 12: Go read this there.
BO’H, 13: empirical does not mean “inference to best explanation or generalisation from a consistent pattern, inductive argument”, so you’re shifting the goalposts, and it is nonsense to suggest that complete enumeration means that something is not empirical – “all swans are white” is a statement that can be tested empirically by looking at all swans (if all swans were white!).
KF, 14: . . . Inferences of inductive character as described are the heart of scientific methods and reasoning. Mathematics, since the days of the ancient Geometers, has been deductive. The number and colour of swans is an indefinite value stretching into the future, the unobserved past and involving unobserved cases in different places — you CANNOT inspect all swans, so inferring whiteness on the pattern was inductive and failed in the 1700’s. A conclusion based on exhaustive inspection of a finite, definite set of cases, is simply not the same as such induction. And note, the issue pivots on reasoning pattern. Mathematics simply does not work in the way Natural Sciences (much less social and psychological/ behavioural ones) do. Indeed, it is the great gap in the naturalistic account, dealing with abstract entities and logical relationships that then by force of how logic affects possibilities and necessities of being, constrain what may be in an instantiated world. The logic of structure and quantity stands athwart the rush of evolutionary materialism and challenges it with cases that cannot be avoided, but whose full significance can be suppressed. For instance, absent a responsible, rational, free, morally governed mind, the integrity of thought and value on truth and doing it right required for Mathematics is fatally undermined. And that brings with it all of that stuff about how can minds be free, and how can minds be morally governed. Where, mere programmed mechanical necessity and/or chance variability do not account for rational, responsible freedom. Cannot account for it. That is where men like Euler speak still, and not just in 0 = 1 + e^i*pi or the like, an expression that shows the deep coherence across huge swathes of Mathematics that in the course of their development did not at all need to come together like that, as far as we know. Which is extremely suggestive on the core nature of the roots of the world.>>
I think this exchange is worth reflecting on. END