I believe that mathematics is not a science. By definition, a science
is a field through which we discover truths about the universe. However,
mathematics at its heart is not to discover how the universe works, but
rather invent tools to be used by scientists in the journey of
discovering truths about the universe. Thus, science discovers truths of
the universe using various rules, most of which are mathematical. Hence,
mathematics is just a tool invented by human to help discover how the
universe works, but what if there are aliens? We know that there are
hundreds of thousands of billions of galaxies in our universe, each of
which has hundreds of thousands of billions of stars, around each of
which there are planets revolving. Hence, there is a reasonable chance
that there are intellectual living organisms other than human. However,
the chance that these aliens have the very same intellect that humans
have is too small. Thus, we can imagine those aliens inventing other
languages and tools to help them understand the mechanism of the
universe and probably still reach the very same truth human scientists
reach through mathematics. Furthermore, they would also probably believe
that Earth and Sun revolve around their common center of mass. However,
it could be the case that 2+3 is encoded in the tools they made with 7,
for example. Therefore, 2+3=5 is not an absolute or universal truth as
mathematics is an invention human made and is not an actual feature of
the universe itself.
For Descartes, I think at least at the beginning he could not see how
2+3 may not equal 5 as he says, “For whether I am awake or asleep, two
and three added together are five, and a square has no more than four
sides. It seems impossible that such transparent truths should incur any
suspicion of being false” (AT 7 20). Then, he goes on and illustrates
why he would doubt the geocentric model of the universe (for instance)
but not that a triangle has three sides (for example) as he says, “So a
reasonable conclusion from this might be that physics, astronomy,
medicine, and all other disciplines which depend on the study of
composite things, are doubtful; while arithmetic, geometry and other
subjects of this kind, which deal only with the simplest and most
general things, regardless of whether they really exist in nature or
not, contain something certain and indubitable” (AT 21 14).
Nevertheless, Descartes believes that everything is finite, but God (if
he exists). Hence, only what could make him question the beliefs that he
asserted were unquestionable is the only infinite thing, God himself (if
he exists). He says, “But what about when I was considering something
very simple and straightforward in arithmetic or geometry, for example
that two and three added together make five, and so on? Did I not see at
least these things clearly enough to affirm their truth? Indeed, the
only reason for my later judgement that they were open to doubt was that
it occurred to me that perhaps some God (if he exists) could have given
me a nature such that I was deceived even in matters which seemed most
evident” (AT 36 25). This was within the period during which Descartes
was assuming that maybe God (if he exists) deceives Descartes and makes
Descartes believes only what God himself wants him to believe in.
Clearly, what God (if he exists) wanted Descartes to believe in is not
necessarily true. Hence, at this point, nothing could ever stand against
the power of God (if he exists) even if it were arithmetic or geometry.
Thus, he doubted arithmetic and that 2+3 necessarily does make 5 as they
may be delusions made by God (if he exists). Afterwards, however, in his
proof of the existence of God, he declared that God is complete,
perfect, and infinite. Hence, God would not like to deceive anyone (at
least not in a bad way). Hence, God does not deceive Descartes, and so
the probability that pushed him to doubt arithmetic has just been proven
wrong. Therefore, after Descartes had showed the existence of God, I do
not think he kept on doubting arithmetic or that 2+3 makes 5.
Personally, I would accept mathematics as being true as the rules
(theorems) of mathematics are not only based on definitions of abstract
objects we invented but also proven using credible philosophical logic.
Additionally, it is evident how beneficial mathematics is to nearly
every single field starting from social sciences to engineering since
thousands of years ago. Unless there is more powerful language that can
be used to reveal the secrets encoded in the universe and help us in
everyday’s life, I genuinely do stand for mathematics.