Thus, in math we reach conclusions deductively, i.e., from universal rules to applications in specific situation/example; e.g., prime numbers are defined by a ‘universally applicable definition’ that determines whether a particular number is a prime or not, and whether a ‘newly discovered number’ is going to be prime. Math follows logic; logic being variously defined but the simplest of all is the Euler’s definition as follows (paraphrased) – It’s is the degree of confidence one has on what one says. Of course, the conclusions will always concur with inductive inferences and experiences/observations.
A simple example of how complete deduction is unique to math is the fact that in step by step math solutions, we need to look only at the previous step to write the current step; all other steps and the ‘question’ itself is irrelevant (in science such is thing is true only for the mathematical steps, but the scientific association with the ‘question’ remains through out its solution to match solution with observational/experiential/experimental inferences).
Similarly, there is only one definition of division that explains 8/2 (the quotient is less than the dividend), 8/1/2 (the quotient is more than the dividend), and 8/2/2 (the quotient is less than the dividend); and what’s 2 3/4 and how it’s so? There also has to be only definition of multiplication for all the three scenario – 2 x 6 = 12 (the product is more than the multiplier and multiplicand), 1/2 x 6 = 3 (the product is more than the multiplier and but less than the multiplicand), 1/2 x 1/2 = 1/4 (the product is less than the multiplier and multiplicand).
Art, on the other hand, is almost totally inductive, i.e., a good piece of art, or music to an extent (music is greatly mathematical in structure but also has leverages to be tempered with), can’t be judged against a set of universal, or even near universal, rules; the features of a piece of art can’t be compared with a standard set of parameters and judged as good, bad, or ugly. In fact, each one of us, has own definition of what’s a great piece of art, at least at the personal level, and any generalisation is once again a personal set of likes. Thus, the way we tend to talk about great pieces of art, or music, is to refer to a disparate collection of them, each liked for a different set of reasons/features.
Science is sandwiched between the two – the induction-intensive art and the deduction-intensive math – thus, in science, specific experiences, observations and experiments count as much as logical connections and thoughts; ‘seeing and feeling’ is very important in science. For instance, ‘internalising’ the concept of weightlessness, for us on earth, is almost an impossible task, because we don’t experience it; but we can more easily realise it if we go through an experience of the same in a simulator. Similarly, the overwhelming majority of us will find it pretty difficult to really understand ‘acceleration’ as opposed to speed. Science is a lot about being out there with all senses sniffing for known and unknown experiences and observations, or creating such experiences in laboratories.
In a way, scientific approach dictates that there may be no place for ‘metaphysical’, overarching truth/reality, and the foundations be laid on observation, experience and experimental findings. Science is about the real world.
Interestingly, math is also about the real world, it’s beyond all levels of elation to accept that mathematical expressions do represent the simplest to the most complex physical, chemical, and now biological realities.