I Eat Rainbows

(Note: this is part of a series of posts where I publish the essays I have had to write for the International Baccalaureate. I might as well get some mileage from them, right? See the index page for more details.)

This essay is released under the GNU General Public License, version 2. Here is a pdf, and source follows.

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%   Copyright 2007 Sohum Banerjea

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title{9) Mathematicians have the concept of rigorous proof, which leads to

knowing something with complete certainty. Consider the extent to which

complete certainty might be achievable in mathematics and at least one

other area of knowledge.}

author{Sohum Banerjea}

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Candidate Code: clq357 par

Session Number: 001161-002 par

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The process of mathematical rigour is the process of checking for

contradictions in a mathematical proof, and if there are none, there

is said to be ``complete certainty'' about the final statement.

This proof, then, is said to be a complete and unshakable guarantee that

the statement in question is true.

This raises a red flag straight away. Proofs cannot, by themselves,

provide complete certainty -- as defined, they need certain premises to work.

Proofs always rely on a body of logic to be the framework for the

proof, and at the simplest, they rely on certain core axioms.

Axioms, statements that are assumed to be true and are then used as the

basis of a logical system like mathematics, are what keeps any proof

from being able to provide complete certainty. Even if the proof merely depends on

``self-evident'' axioms, there is always the possibility that these axioms

are invalid, removing the absolute-ness of the absolute proof.

Descartes, a 17th century philosopher, attempted to create a

philosophy based on merely one premise: ``I think; That which thinks

must exist, thus, I exist.'', commonly abbreviated to ``I think,

therefore I am.''

But even this philosophy depends on the assumptions ``That which thinks,

must exist'', and ``I think''. It relies on these being true, and

although we dismiss these as irrelevant, it is still a dependence, and thus

still prevents the philosophy in question from complete certainty.

So, it is not possible to have an ``absolute proof''. However, it is

possible to have an absolute proof, assuming certain axioms are

true. Since this is a more meaningful definition, I shall be using it in

the rest of the essay.

Then, the next point is, which set of axioms? In mathematics, this may

seem to be a simple question -- just use the axioms that are used

``normally'' -- that is, use the axioms that give rise to the number

system we have today, like the Axiom of Induction.

However, this is not as simple as it seems. G"{o}del showed us that

all consistent mathematical systems are either non-computable or

incomplete.

What this means is that it is impossible to formulate a system of

mathematics, a set of axioms, which satisfies two separate criteria.

These are a) its theorem space is computable -- that is, a given a

sufficiently long, finite, amount of time, all the possible theorems

in it can be calculated, and b) that the system of mathematics is

powerful enough to prove or disprove every possible statement that can

be formulated within it -- that is, give them a ``truth'' in

the context of the axioms. cite{godelpaper}

Thus, although most mathematical statements can be given an

``absolute'' validity, where ``absolute'' is dependent on the axioms

used, there will always be some statement within any reasonable,

usable mathematical system which cannot be absolutely shown to be ``true''.

A point against this is that, since G"{o}del's theorem has to be

formulated in a logical system, it can only be as true as emph{its}

axioms.

This is true. However, G"{o}del formulated his theorem in first order

logic (FOL) cite{godelpaper} -- a system that formally mimics the way our mind works. If the axioms of

FOL are not true, that would mean the very processes our

brains utilise to reason are flawed. Then we would have no way to even

find out that FOL is flawed, since we would have to

reason about it with the implementation of FOL in our brains.

This leads us to a rather unique situation. Mathematics, the most removed and

abstract of systems, has been let down in one specific instance -- all

that is required -- by the fallibility of the human brain. As far as

humans are concerned, FOL emph{is} an absolute truth.

Adding to the confusion are the different possible mathematical systems we

could use that differ in areas far removed from  ``everyday''

mathematics, including surreal numbers cite{knuthsurreal},

which implement the concept of infinitesimals (roughly the difference

between $0.dot{9}$ and $1$, which in ``normal'' mathematics is

nothing).

There are many different possible mathematical systems, and all of them

have equal claim to being the ``true'' one, if there is indeed such a

thing. The common subset of some of them that we use in day-to-day

mathematics has not been chosen for any ``true-ness'', but because it is

simple, easy to grasp, and useful.

So, the concept of ``complete certainty'' is unclear even in the most logical

and abstract of fields, that of mathematics. If this is so, what hope do

other, more real-world applicable fields, fields that are burdened with the physical

world that mathematics tries to abstract away, have?

Surprisingly, there exists a roughly analogous amount. For although

the same real-world problems that plague mathematicians in search of ``complete

certainty'' apply to scientists as well, and introduce an

``as-far-as-we-are-concerned absolute'' into the equation, they also

manage to ground science, and base it firmly in reality.

This may be an odd statement to make, as we are used to thinking about

science in terms of hypotheses and proofs. If I make a hypothesis, and

then I collect data which matches it, surely the hypothesis in

question is proven?

That is not exactly correct. It commonly said that ``A

hypothesis cannot be proven, merely disproven.'' This is because

testing a hypothesis involves getting real-world results that support

it, but there is no way of guaranteeing that that the emph{next

result will not be different.}

The other thing that we cannot guarantee is whether the hypothesis is

only an approximation to the truth. The classical example is Newton's

Laws of Motion, and Einstein's Relativity. Under the

situations in which the Laws of Motion were tested, they were perfect,

but it was later shown that they are merely an approximation to

Special Relativity, and break down when the speeds of the objects in

question are an appreciable fraction of the speed of light.

cite{einsteinrelativity} Relativity

today breaks down when trying to explain the behaviour of very small

particles, and in this realm it has been superseded by quantum mechanics.

We use relativity and quantum mechanics now, not because they are the

``truth'', or because they have been ``proven'', but because they are

the best approximation to the truth that we have. cite{myth}

Because science has to fit everything it generates to the real world,

including our potential absolute truths, this guarantees that the

statements in question at least have an element of truth in them, that,

hopefully, grows as the scientific method is further applied. For

instance, the world was originally ``known'' to be flat, then a

sphere, and today an oblate spheroid, each of these being closer and

more successful approximations to ``the truth''.

cite{relativityofwrong}

Of course, this does assume that our senses are infallible -- for, after all,

what else are we observing the real world to match the hypotheses with?

If our senses are fallible -- and we have ample evidence of this, including

instances where two people have observed the same event contradictorily,

usually explained as one person ``hallucinating'', then we cannot be

sure that the ``real world'' that we sense and then match to scientific

hypotheses is the real world. To paraphrase a movie, ``The

Matrix'', my hypothesis that there exists a spoon in the area in front of

me at this point in time may be incorrect, although my sensing of the

world around me would seem to confirm it. My

senses can be fooled.

Following this line of thinking to its logical conclusions, one

reaches

radical, if not very interesting, ideas. What is to say that you, the

person whom you perceive as thinking and thus existing, is not completely

fooled? What makes you certain that you are not a construct in a computer,

or a brain floating in a scientist's vat and being stimulated extremely

precisely?

The reason these ideas are not very interesting is a point that is very

important when searching for the absolute truth in science. Ultimately,

emph{it does not matter}. It does not matter to me whether the world I

perceive with my senses is the real world or not, because, by definition, I

have no way to sense this difference, and thus there is no work I can do to

deduce the absolute truth of this actual ``real world''. It does not impact

us, and thus can be ignored.

Thus, it is then theoretically possible to reach an absolute truth in science, even

though we might never reach it, as long as we remember that this ``absolute

truth'' may not be the actual ``absolute truth'', but merely the truth of

some construct in a larger universe which we cannot perceive.

Thus, both disciplines return to human fallibility producing an

obstacle to the search for the absolute truth. It is possible to find

emph{the} absolute truth, theoretically, in both disciplines. However,

because of human fallibility, we bump against a barrier. We can find

emph{an} absolute truth in both disciplines, but only as far as our

physical limitations will allow us to do so.

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begin{thebibliography}{20}

bibitem[Newman (2005)]{descartes}Newman, L. 2005.

underline{Descartes' Epistemology}. Stanford

Encyclopedia of Philosophy. [Internet].

Available:url{http://plato.stanford.edu/entries/descartes-epistemology/}

[Accessed 6 August 2007]

bibitem[Asimov (1989)]{relativityofwrong}Asimov I. The

Relativity of Wrong. ul{The Skeptical

Enquirer}. [Online]. vol.14.

Available:url{http://chem.tufts.edu/AnswersInScience/RelativityofWrong.htm}

[Accessed 18 August 2007]

bibitem[Hofstadter (1979)]{gebegb} Hofstadter, D. 1979.

ul{G"{o}del, Escher, Bach: An Eternal Golden

Braid}. New York: Basic Books.

bibitem[G"{o}del (1962)]{godelpaper} G"{o}del, K. 1962.

ul{On formally undecidable propositions of

Principia Mathematica and related systems.} Dover.

bibitem[Knuth (1974)]{knuthsurreal} Knuth, D. 1974.

ul{Surreal numbers: How two ex-students turned

on to pure mathematics and found total happiness}.

Reading, Massachusetts: Addison-Wesley.

bibitem[Einstein (1924)]{einsteinrelativity} Einstein A.

1924.

ul{Relativity: The Special and general

Theory}. Methuen & Co Ltd.

bibitem[Horner and Rubba (1978)]{myth} J Homer, P Rubba.

1978.

ul{The Myth of Absolute Truth}. The Science

Teacher.

end{thebibliography}

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