Shankar Bhattacharyya wrote:
Absolute proofs exist only in mathematics and in formal logic, but
not in the natural sciences.

Would you actually claim that that is really true of mathematics and
formal logic?

Mathematics depends on a set of axioms. Axioms are "obvious
assumptions". Formal logic provides the tools of inference by which we
extend axioms into theorems and evaluate information for consistency,
implication, so on so forth.

The absolute proofs of mathematics depend on the assumptions.

In some sense, in possibly some situations, maybe in many, the proofs
may not depend on the assumptions, in a mechanistic sense. However, what
the proofs prove presumably does depend on the assumptions. Otherwise we
would not need axioms.

I think the crux of the matter is not whether assumptions are made.
Rather, I see the difference as follows. In the natural sciences, there
is in some sense a set of factual truths, which we seek to discover.
Because nature is under no constraint to make these truths follow any
specific pattern, we can never "prove" any such pattern. We can only
assume those patterns hold until countervailing evidence appears.

In mathematics, on the other hand, we humans are in a more creative
position. We define the axioms. They are not "out there" for us to
find, although we may attempt to define them in such a way that they
model some facet of reality. For example, we can either assume the
Axiom of Choice, or not. Either way is valid, but our decision has a
definite and tangible impact on which things we can represent in our
theory, and which things we cannot.

Brian Tung
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