There
are three key differing beliefs in mathematics when discussing
whether a mathematical proposition is in any way related to truth or
reality: Platonism, Formalism and Intuitionism. These perspectives
are actually applicable to language in general. However, all a
mathematical proposition really is, is a statement. “One apple plus
one apple equals two apples.” is the same as “1+1=2”
Plato
believed that there was a distinction between what we perceive to be
our reality and what reality is.
He argued that we should take our beliefs and analyse these, reducing
and questioning our assumptions until we reach an ultimate truth or
reality (the dialectic).
He
distinguished between the one and the many. For example, there is a
potential infinity of things we call gold in the world: gold
necklaces, gold rings, gold paper, and so on. However when we ask
what gold is, it is the element Au,
it is what all these things have in common.
Mathematics
seems to be a universal language. For example, say you have an
Englishman and a Frenchman and they both walk into a bar…. Or you
know are just chatting in any other location. The Frenchman asks the
Englishman what he means by “cheeky”. There is no French
alternative that completely embodies the meaning of the word. However
if there are circular beer mats on the table, cylindrical cheeky
glasses of beer on the bar, and some round bar stools outside, and so
on, they will both recognise the shape of the circle. They will both
know that the connection between these things is a circle. Plato
believes that this is because we recognise the true form of the
circle, and that this true form is the reality rather than just our
perception of many objects that are circular. There can hardly be a
perfect circle ever created in our physical universe due to the
nature of pi, but the concept of what a circle is, is known to many.
It is important to note that Plato did not think that these true
forms were ideas but believed that these are external and objective
to us, and even to space and time.
Platonism
within mathematics takes his ideas and argues that there are abstract
mathematical objects whose existences are independent of us. The
mathematical objects have true forms, not just the circle as per the
above but sets, equations, propositions, and so on. Mathematical
truths are therefore discovered using the dialectic, rather than
being invented. If you look at this position from a mathematical
perspective it is quite convenient. Firstly we could potentially
solve any mathematical problem, as we only have to discover
rationally all the mathematically true forms that could exist.
One
of the problems with the Platonist argument is that if these truths
are completely independent to us, how do we test these ideas to see
if they are true? Take the gold example. If I say “this necklace is
gold” we can test this by analyzing the necklace and confirming
that it is or is not made of Au.
However with Plato’s viewpoint it seems that we will always
perceive reality differently from its true form. So if we test the
proposition “One plus one equals two” even though our perception
would be that this is correct, if these are abstract and independent
from our perception then how is this useful? One of the reasons
mathematics is so important to our views of existence is that it
enhances scientific theories. However it seems that by claiming
mathematical objects and propositions are independent of our
perceptions, Platonism disregards this altogether.
Intuitionism
argues that instead of having these independent forms, mathematics is
just a creation of the mind. Mathematical propositions can only be
proved true by reasoning that proves it to be true – and therefore
we can communicate mathematics only if other minds have come to the
same logical conclusion. We can rationally postulate mathematics, but
it can also be applied to every day empirical reality if you believe
that the mind and body are interlinked – therefore corresponding
with science.
How/why
do we have maths in the first place? Where did this thing that so
many people struggle with/become super geniuses at come from? Look at
your hands! How many fingers are you holding up? How many slices of
cake have you eaten today? How much money do you need to give the
shopkeeper for that pinot grigio and packet of fags? Maths stems from
counting, counting became measuring for house building and such
(remember back to your school days the endless bore of Pythagoras'
theorem?) With Platonism all of this is irrelevant. However with
Intuitionism you could argue that although these empirical things are
not the
mathematics,
the logic and reason we have used to create them in our minds is.
The
problem with this theory is when we introduce mathematical entities
called irrational numbers. Remember that thing called Pi (π)
(mmm pie). You used it to calculate the area / circumference /
diameter of a circle – maybe you still do. But what is Pi? Pi is a
really long number beginning with 3.14159265358… The number of
decimal digits on this number is infinite, but unlike rational
numbers they form no pattern. This means that even if we manage to
calculate the next digit, we will have no idea what the one after
that is. The problem with the irrational number for intuitionism is
that it is irrational. Therefore we can’t just derive this in our
minds. For Platonism this would not be a problem – for example
where Intuitionists would argue that infinity or an irrational number
could not exist as we have no experience of this in our physical
world, Platonists would argue that there was a true form of infinity
or of pi we are yet to discover.
So
if these two arguments don’t work is maths just devoid of meaning?
If we are not really sure if these independent abstract ideas exist
and we can’t just make them up in our heads, what is a mathematical
proposition and how does it work? Formalists argue that mathematical
propositions are just a game we play, making up a story. Mathematical
propositions and concepts are part of the story of maths in the same
way the tardis is part of the story of Doctor Who. However just like
a story these things make sense in the story but not outside it.
There is a man who flies through time and space in a police box…
that doesn’t make sense in the same way that 'what the hell does
10/2=5 actually mean?' does. Unlike sciences such as biology which is
a study of something else – life, maths is just the study of maths.
Mathematics studies quantity, structure, space and change for
example, but these are mathematical concepts themselves. However,
unlike Doctor Who, mathematics is logical and we can use it to
describe objective things outside of mathematics. For example we can
use it to model and predict the weather – but the weather isn’t
actually mathematics – we’ve just made it mathsy by putting our
perspective on it.
So
are people who claim that mathematics is on a higher plane of
existence talking out of their bums? I think that the only answer can
be sort of, sometimes and depends. I’ve only written about a few of
the theories here and in that not gone into much detail. However all
of them relate to what you think quantifies as existence in the first
place – is existence just in the mind or do you believe that
there’s something else out there, and if so which is the higher
plane? Platonism depends a lot on faith that these objects exist -
but would you say that someone who has a belief in religion,
philosophy, and so on, was a bum talker? If maths is just a game,
but can be useful, does this mean it’s any less part of existence?
Rhiannon
Whiting
