Re(Axiomatics)
Andrei Heilper
heilper.andrei at gmail.com
Sun Apr 26 19:43:21 EDT 2020
Dear list subscribers.
I am (and I imagine other subscribers are also) confused by the discussion
about axiomatics.
Does axiomatics refer only to formal systems or we include here the
deductive method in general?
Do we refer to the process by which we choose our axioms or our undefined
notions?
Let me address some points raised. Claim like that Lobachevsky did
not work within axiomatics is IMHO wrong. And in what way the parallel
problem stopped the development of mathematics?
Abstraction exists even when we are not aware of it. For example, the curve
describing the boundary of your garden does have a zero width - otherwise,
you will have to talk not with your neighbor but with his lawyer.
Physics was not entirely formalized by Newton and even today it is
difficult to present it as a formal system (albeit Suppes' work is a
good source which also was considered highly controversial).
There was a claim that mathematics should ignore axiomatics and base itself
on the existing corpus of mathematics. Then the question arises how this
corpus was obtained.
Category theory and universal algebra are constructions that do not depart
in any way from axiomatics.
Should philosopher's axiomatics be different from that of mathematics? If
so then what is the difference? (IMHO Spinoza's Ethics is not a really an
axiomatic system at all).
Archimedes indeed used analogies that were not belonging to an axiomatic
system but used the principle of the lever (which on the other hand he
proved). But the final result was obtained by formal proof.
And finally, let me go down to the basics. Among the first theorems proved
and credited to Thales is that the opposite angles of two intersecting
lines are equal. There are some historical sources that the Egyptian rope
stretchers were not sure that these angles are equal and verified this by
measurement.
Should we collect this and all mathematical knowledge as empirical facts
and give up the most powerful tool invented by humankind? Then probably the
science which is called today mathematics can be removed from the
curriculum for the joy of many.
Andrei Heilper
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