Friday, March 9, 2012

The Objects of study in mathematics are discovered, not invented.


Is Mathematics Invented or Discovered?
Why math is essentially not invented by humans.

Introduction
I seek to demonstrate why math is essentially not an invention of humans by providing my own arguments.  I will clearly state what my assumptions are and warmly encourage feedback; one beautiful thing about this topic is that one needn't be an expert mathematician to have a well-thought-out opinion on the subject.


Mathematics
To kick things off, I should state what the definitions are for the words mathematics and invention.  Even mathematicians might disagree on what the definition of mathematics is.  According to [1], mathematics has been defined in various ways:

Mathematics is the study of quantity, structure, space, and change.  Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.

David Hilbert defined mathematics as follows: a conceptual system possessing internal necessity that can only be so and by no means otherwise.

I want to underscore the distinction between what is called mathematics and the objects of study in math.  Roughly speaking, the former is an invention of many humans.  An important example of what humans did invent is numeration [2].  We have a system of names and notations that has been invented by many humans over the years for things like 0, 1, 2, etc.  Since these numbers will come up later, I will say now that the totality of all such numbers is called the set of natural numbers (though some people exclude 0 from that list, mainly for technical reasons).

However, our system of names and notations is not what the names and notations refer to.  In the same way that a map is not the territory, the system of names and notations which some might in itself call mathematics is not what those names and notations reference.  What is referenced by the system are what I call the objects of study of math.  There is an important difference between the abstract concepts (the territory) and their names and notations (the map).

With this in mind, allow me to refine what I will demonstrate:

The objects of study in mathematics are essentially not invented by humans.

Objects of Study in Mathematics
Let me take a detour to explore what the objects of study in mathematics are.  For that, certain things need to be mention such as axioms, the notion of consequence closure, and statements.

Statements
For the more technically-oriented reader, when I say statement, you can intuitively take that to mean a well-formed formula [3].  Otherwise, a statement is a grammatically-correct utterance in a language.  For some statements, it makes sense to ask if they are true or false.  The sort of statements we need only concern ourselves with at the moment are mathematically-oriented statements, those which could be viewed as well-formed formulas.  Here are some examples of statements along the lines of what is relevant:

0=0

0=1

1+1=2

2+2=5

The number zero exists.

The empty set exists.

The set of natural numbers exists.

If a, b, and c are the lengths of the legs in a right triangle where c is the largest of the three, then the square of c equals the sum of the squares of a and b.

Two triangles such that corresponding side-angle-side's are congruent are congruent.

There is a set that is an element of itself.

There is no set that is an element of itself.


As these examples illustrate, statements as I am defining them here are utterances for which it makes sense to ask "is it true".  Some of the above statements are false and some are not false.

Axioms
An axiom is a special kind of statement.  Though an axiom is commonly defined to be something which is self-evident, true and not requiring proof that it is true, the way I think about axioms is that they are assumptions made for the sake of argument.  Let's jump into two important examples involving axioms: Euclidean geometry and Peano arithmetic. 

Here are a couple of the five axioms of Euclidean geometry followed by a few of the nine axioms of Peano arithmetic:
·         All right angles are equal to one another.
·         (Parallel postulate) If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
·         0 is a natural number.
·         For every natural number n, the successor of n (which is n+1) is a natural number.
·         (Induction principle) Let K be a set containing the natural number 0 with the property that the successor of n is an element of K whenever n is an element of K.  Then K contains all natural numbers.

Axioms are not considered absolutely, unconditionally true.  Rather, one temporarily assumes they are true for the sake of argument for working within the scope of the axioms.  When you are working in Euclidean geometry, you will constantly assume the five axioms of that system.  When you look at non-Euclidean geometry (used in Einstein's theory of relativity), in which various negations of the parallel postulate are employed, you no longer consider the parallel postulate to be absolutely, unconditionally true.  Instead, you assume--again, for the sake of argument--that some negation of the parallel postulate is true in order to make your arguments about non-Euclidean geometry valid.  Axioms can also be said to be the assumptions needed to make an argument valid, in line with David Hilbert's definition of mathematics.

Consequence Closures
The consequence closure of a set of statements (or well-formed formulas, if you like) is another set of statements, sometimes called theorems, which are entailed by the original statements.  An important example is when the original set of statements is a collection of axioms.  The consequence closure of the axioms of Euclidean geometry is the set of all theorems in that geometry which follow from (i.e., are entailed by) the five axioms.  One can say that the consequence closure of the axioms of Euclidean geometry literally is Euclidean geometry.  One can say that the consequence closure of the Peano axioms is Peano arithmetic.

A crucial example of a consequence closure is that which is known as Zermelo–Fraenkel set theory (a.k.a. ZF set theory).  In the link, eight axioms are stated.  ZF set theory is the consequence closure of that collection of axioms.  A set theorist is the one charged with the task of determining whether a given statement is in that consequence closure or not which is to say to determine whether a given statement follows from the axioms of ZF set theory.  (I'm skipping over the concept of axiom schema which are really themselves just collections of axioms.)

My final example of consequence closure will be something much easier to wrap our heads around: the consequence closure of the set devoid of all statements, i.e., the consequence closure of the empty set. This consequence closure equates to the set of all statements which are "self-entailed," namely the set of all tautologies [7].  There are three tautologies which are distinguished in Aristotle's logic:
1.      the law of the excluded middle: the disjunction of a statement and its negation is true
2.      the law of identity: a thing is itself
3.      the law of non-contradiction: it is not the case that the conjunction of a statement and its negation is true.

Objects of Study in Math (revisited)
Consequence closures are the objects of study in mathematics.  A set theorist studies the nature of the consequence closure of set-theoretical axioms.  A geometer studies the nature of the consequence closure of geometrical axioms.  A topologist studies the nature of the consequence closure of the axioms of topology.  And so on.  These are just types of mathematicians.  All mathematicians study the nature of various consequence closures. 

As an aside, I should add at this point that the notion of consequence closure is a concept delved into in metamathematics (a.k.a. mathematical logic) which is the study of math itself.  It may be the case that most mathematicians have never heard of the term consequence closure but their work is still the study of the nature of various consequence closures nonetheless.

Invention
To invent means to create, originate, and fabricate.  When someone says "X is invented," that implies that X did not exist until then.  Thus the statement "X is invented" needs all of the following to be true: (a) X did not exist until then, (b) X exists after then, (c) X owes its existence to the process of invention, perhaps as carried out by a human.  To rephrase (c): X exists because of its inventor (human or otherwise).

Why math is essentially not invented by humans
Finally, I am in a position to provide an argument for the main conclusion of this article which is that the objects of study in math are essentially not invented by humans.

The first step in my proof is to show that the set of natural numbers was not invented by humans, i.e., the set of natural numbers has ontological primacy with respect to humans.  I will employ two different arguments to establish this.
Here I will assume for the moment that humans will eventually cease to be (as a species).  I will also use the fact that there is no largest natural number.  Suppose that the set of natural numbers depends on humans to exist. The dependency entails that a number exists if and only if it is invented by a human.  If humans will eventually cease to be, that means there is a fixed duration during which humans have the opportunity to invent natural numbers and, consequently, a largest natural number.  Since there is no largest natural number, the original supposition that the set of natural numbers depends on humans to exist is wrong; i.e., the set of natural numbers was not invented by humans.

To take another approach, let me argue that Peano did not invent the set of natural numbers and, by extension, without loss of generality, no human invented it.  One could cite the exact time and date that Peano wrote the nine axioms of Peano arithmetic and suggest that that is when the set of natural numbers came into being.  Consider those nine axioms, listed in [5] (and I acknowledge that Peano's original thesis might not have been formulated precisely as it is written in [5]).  Those particular nine statements, while perhaps first written in total by Peano, existed prior to him writing them down.  This can be seen by imagining that all written data is erased from the universe.  The day after the erasure, I could come along and write down those nine statements, so they still exist even without written proof that they exist.  This is still the case the day prior to Peano writing them down, and is still the case prior to anyone writing them down.  (Circumstantial evidence that natural numbers existed prior to people like Peano is that there certainly were things to count prior to humans being around to count them and humans are not the only types of things which can count.)  Since the nine axioms of Peano arithmetic, in total, which describe the set of natural numbers, existed before Peano wrote them down, part (a) of my definition of invention is violated.   This shows that the set of natural numbers was not invented by Peano.  This argument applies equally well to all mathematicians, showing that no mathematician (human and not) invented the set of natural numbers.


Next, I want to prove that the set of statements was not invented by humans. 
There are two alternate arguments I will give you, each with a different starting assumption.  This is not a circular argument because I only said that the nine axioms of Peano arithmetic existed prior to any human writing them down.
(a) (Assumption 1) All sets have equal ontological primacy.  If all sets have equal ontological primacy, then the set of statements has the same ontological primacy as the set of natural numbers.  By an argument above, the set of natural numbers does have ontological primacy with respect to humans, so the set of all statements does also.

Can we prove that all sets have equal ontological primacy?

(b) (Alternate assumption) A set that can be encoded by natural numbers has the same ontological primacy as the set of natural numbers.  Using Gödel-numbering [11], the set of all statements can be encoded by natural numbers.  Using the assumption, the set of all statements has the same ontological primacy as the set of natural numbers.  By a previous argument, that means that the set of all statements has ontological primacy with respect to humans, i.e., the set of all statements was not invented by humans.


Putting these notions together, the objects of study in math (consequence closures of sets of axioms) are not invented.
The consequence closure of a set of axioms is not a human invention.  Recall that the consequence closure of a set of axioms is the set of all statements entailed by that set of axioms.  Given the infinite number of them, it is usually impossible to write down all statements entailed by a set of axioms.  A human couldn't have invented the set of all statements entailed by a set of axioms, given our limited time and the infinite length of the set of consequences.  To see why the consequence closure of a set of statements is usually infinite, just note that if S is a statement that is a consequence, then the statements, "not not S," "not not not not S," "not not not not not not S," etc., are also consequences and this process can be done indefinitely.

The core argument is this: given that no statements were invented, and that consequence closures were not invented, the objects of study in math were not (and are not) invented.

Conclusion
The following quote is attributed to Michelangelo: "every block of stone has a statue inside it and it is the task of the sculptor to discover it."  In the way that a statue is discovered, a mathematical object is discovered.  Central to the reason behind this is that a statement is likewise discovered, selected really, amongst the sea of possible statements and the resulting mathematical object, such as set theory or arithmetic, exists independently of humans.




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References and "see also"
[1] http://en.wikipedia.org/wiki/Mathematics
[2] http://en.wikipedia.org/wiki/Numeration
[3] http://en.wikipedia.org/wiki/Well-formed_formula
[4] http://en.wikipedia.org/wiki/Euclidean_geometry
[5] http://en.wikipedia.org/wiki/Peano_arithmetic
[6] http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
[7] http://en.wikipedia.org/wiki/Tautology_%28logic%29
[8] http://en.wikipedia.org/wiki/Laws_of_thought
[9] http://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem
[10] http://en.wikipedia.org/wiki/Natural_numbers#A_standard_construction
[11] http://en.wikipedia.org/wiki/Godel_numbering

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