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@ -5,9 +5,9 @@ date: 2019-03-18
* Introduction
I have recently bought the book /Category Theory/ from Steve
Awodey [[ref-1][(1)]] (which is awesome, but probably the topic for another
post), and a particular passage excited my curiosity:
I have recently bought the book /Category Theory/ from Steve Awodey
citep:awodeyCategoryTheory2010 is awesome, but probably the topic
for another post), and a particular passage excited my curiosity:
#+begin_quote
Let us begin by distinguishing between the following things:
@ -54,8 +54,9 @@ date: 2019-03-18
this subject.
In this post, I will try to share my path through Peano's axioms
[[ref-2][(2)]], because they are very simple, and it is easy to uncover basic
algebraic structure from them.
citep:gowersPrincetonCompanionMathematics2010, because they are very
simple, and it is easy to uncover basic algebraic structure from
them.
* The Axioms
@ -188,29 +189,21 @@ date: 2019-03-18
It is remarkable how very few (and very simple, as far as you would
consider the induction axiom "simple") axioms are enough to build an
entire theory of mathematics. This sort of things makes me agree
with Eugene Wigner [[ref-3][(3)]] when he says that "mathematics is the science
of skillful operations with concepts and rules invented just for
this purpose". We drew some arbitrary rules out of thin air, and
derived countless properties and theorems from them, basically for
our own enjoyment. (As Wigner would say, it is /incredible/ that any
of these fanciful inventions coming out of nowhere turned out to be
even remotely useful.) Mathematics is done mainly for the
mathematician's own pleasure!
with Eugene Wigner
citep:wignerUnreasonableEffectivenessMathematics1990 when he says
that "mathematics is the science of skillful operations with
concepts and rules invented just for this purpose". We drew some
arbitrary rules out of thin air, and derived countless properties
and theorems from them, basically for our own enjoyment. (As Wigner
would say, it is /incredible/ that any of these fanciful inventions
coming out of nowhere turned out to be even remotely useful.)
Mathematics is done mainly for the mathematician's own pleasure!
#+begin_quote
Mathematics cannot be defined without acknowledging its most obvious
feature: namely, that it is interesting --- M. Polanyi [[ref-3][(3)]]
feature: namely, that it is interesting --- M. Polanyi
citep:wignerUnreasonableEffectivenessMathematics1990
#+end_quote
* References
1. <<ref-1>>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic
Guides 52. Oxford ; New York: Oxford University Press, 2010.
2. <<ref-2>>Gowers, Timothy, June Barrow-Green, and Imre Leader. The
Princeton Companion to Mathematics. Princeton University
Press, 2010.
3. <<ref-3>>Wigner, Eugene P. ‘The Unreasonable Effectiveness of
Mathematics in the Natural Sciences’. In Mathematics and Science,
by Ronald E Mickens, 291–306. World
Scientific, 1990. https://doi.org/10.1142/9789814503488_0018.