Nature and Construction of Nature and Construction of Knowledge in MathematicsKnowledge in Mathematics

What mathematics is all about, how it came about,

and why is it that it is irrefutable

What is mathematics?• Mathematics is not about answers, it's about processes.

• For more than two thousand years, mathematics has been a part of the human search for understanding. Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning. These remain fruitful and important motivations for mathematical thinking, but in the last century mathematics has been successfully applied to many other aspects of the human world: voting trends in politics, the dating of ancient artifacts, the analysis of automobile traffic patterns, and long-term strategies for the sustainable harvest of deciduous forests, to mention a few. Today, mathematics as a mode of thought and expression is more valuable than ever before. Learning to think in mathematical terms is an essential part of becoming a liberally educated person.                                   -- Kenyon College Math Department Web Page

What is mathematics?• It is one of the most flexible subjects. It is one that forms the basis of

civilization as we know it.

• It is obviously used in subjects as varied as history (dates, years, number of whatever…), as well as physics (need I say?)

• Math, in a way, is a logic and reasoning based language, that gives us a set of rules and codes to write and communicate logic, as well as to derive conclusions from it.

• It is a precise form of expression, it is not possible to misinterpret what math has to say, if you know what the rules define.

• As opposed to physics or chemistry, theorems in math do not assume anything. That is to say, they arrive at new sets of conclusions, through careful reasoning and within the parameters of logic.

(I’ll be coming back to this part…)

How did it all begin?

• It all started with EuclidEuclid• Mathematics itself existed

before him too, but his greatest accomplishment was to present them in a single, logically coherent framework, including a system of rigorous proofs that remains the basis of mathematics 23 centuries later.

Depiction of Euclid – Artists impressionNo real description of Euclid survives

So what did the guy do?

• Euclid based mathematics on “axioms”. These are fundamental basis of all reasoning. They are truths that do not require proof, and are universally accepted.

• One may argue, that without proof, the “argument” becomes weak, however, the axioms are so basic, that challenging them is tomfoolery at best. This guy doesn’t look like the guy in

the first pic, right?OBVIOUSLY!

No one knows what he looked like in the first place!

Tell me what these axioms are!!

• The statements – "any two points can be

joined by a straight line segment" (fig 1)

– "any straight line segment can be extended indefinitely" (fig 2),

– “Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center” (fig 3),

are three of these axioms.

der Herr Warum – “Mr. Why?”

• Several questions regarding the merit of mathematics (since it is based on axioms) have been raised.

• The most pressing one being – – if you have a set of axioms

describing a mathematical system, do the rules for logical reasoning which they gave in their book allow you to derive every true statement about the system, and do they ensure that only true statements can be derived?

der Herr Warum – “Mr. Why?”

• The obvious answer is “yes”.

• A certain Kurt Gödel (called “Mr. Why” for his inquisitiveness) confirmed this.

• His dissertation established that the principles of logic developed up to that time were adequate for their intended purpose.

BUT …

• However, Gödel has also presented the Incompleteness Theorem. He said that that there will always remain some statement (or natural numbers ) that will remain outside the purvey of these rules and axioms.

• Although Gödel's work irrefutably proves that "undecidable" statements do exist within number theory, not many examples of such statements have been found.

Example- • One example comes from the sentence:

This statement is unprovable. You can see why this is a prime candidate:

- If you could prove this statement to be true, then it would be false!

- It is true only if it is unprovable, and unprovable only if it is true!!

As it stands, this is not a statement about the natural numbers. But Gödel had devised an ingenious way to assign numbers to English-language phrases like this one, so that finding whether the statement is true or not translates to solving numerical equations. He proved that, within the axioms of number theory, it is impossible to prove whether or not the equation corresponding to the sentence above holds true, thus confirming our "common-sense" analysis.

So what’s the point?

• Actually, I don’t know… but here’s what it can mean –

• There are things that are SEMANTICALLY true in mathematics that cannot be proved by any finite axiom system.

• Another consequence of the theorem is that any system that can prove its own consistency is inconsistent.

• I am as confused as you… the theorem is still under deliberation, so its okay for now.

May Gödel’s Theorem Rest in peace…Moving on …

• There exist several schools of thought. In reality, the debate on whether mathematics is truly a precise and complete science has been going on for quite some time.

• Due to this there are many schools of thought in mathematics. It is important to note, that all the schools look at mathematics in a very different light, and hence shape the very application, perception, and meaning of mathematics.

LOGICISM

In the philosophy of mathematics, the thesis that all mathematical propositions are expressible as or derivable from the

propositions of pure logic.

PLATONISM

This school holds that mathematical concepts exist independent of any human

realization of them.

INTUITIONISM

This philosophy holds that only those mathematical concepts that can be

demonstrated, or constructed, following a finite number of steps are legitimate.

FORMALISM

This one’s a bit complicated – so PAY ATTENTION.

• In mathematics, the mode of thought that everything can be incorporated into a formal system (with axioms, well-formed-formulae, grammar, and rules) and can be formally discussed and deliberated within the formal system.

• Formalism holds that mathematics consists simply of the manipulation of finite configurations of symbols according to prescribed rules; a “game” independent of any physical interpretation of the symbols.

Conclusion (just for the schools of thought… don’t feel happy

just yet)

Everything comes out of pure logic

Formal system, formal rules. That’s all we need to formally know everything

Mathematics exist. Whether people realize it or not, is immaterial.

Anything that is, can be proven using legitimate logic, in a finite number of steps.

Now the real question - Now the real question - Why do we need math in the first Why do we need math in the first

place???place???

Math as a subject is used in everyday lives of everyone. The

following slides serve as an example -

RELEVANCE – what is mathematics to the lay man? How does he use it to construct his own understanding of his surroundings? How is math important in every day activities?

• Suppose you are building a small bridge -

Now the real question - Now the real question - Why do we need math in the first Why do we need math in the first

place???place???

You know how long its going to be …You know how long its going to be …

You know how high its going to be …You know how high its going to be …

But you need to know how much string is needed to hold the thing up!!But you need to know how much string is needed to hold the thing up!!

The answer is simple – use the PYTHAGORAS THEOREM!The answer is simple – use the PYTHAGORAS THEOREM!

x

Z2 = x2 + y2

y