A Panoramic view of Theoretical Mathematics

A Panoramic view of Theoretical Mathematics

Theoretical Math is really fun to study. One is often required to balance intuition and mathematical rigor. Both are required and equally essential. While an applied Math student can avoid lot of questions, a theoretical math student is someone who has an innate need to understand the internals of Mathematics.

If someone is satisfied in using e^x for x = a real number [perhaps calculating it and using it in other equations] and is not confounded with any fundamental questions , then one is probably better as an applied math student. But for a theoretical math student this expression e^x is a very profound expression -- what is "e", what is "x" and what is a real number ? Lot of interesting theorems and ideas behind all this. Here is a panoramic view of how it appears when one tries to peek into the fundamentals....

1. Natural Numbers :

The story has to start with natural numbers. They appear simple , just counting numbers. Well, that is not enough, one needs to define these precisely. Obviously to say 1,2,3 ... is not ok. One needs a very precise, rigorous definition with minimal assumptions. What if one uses Φ, {Φ}, {Φ,{Φ}}, ... instead of 1,2,3 [with Φ being some symbol or the empty set]?

Well, welcome to peano axioms and the wonderful world of mathematical induction. Perhaps one can spend a whole career here -- include a little set theory and study the cardinal aspects of natural numbers also and lo! one can go on and on proving what is a set and what is not ... why 3 != 4 etc. And if one does not get stuck here , and decides to move forward ...

2. Integers :

Next one needs to learn about integers. I can imagine an ideal mathematician who ended up with natural numbers and never came out of the natural number world– but am yet to find one. Lets move to integers now. So, we need this operation called subtraction. and as children we just learnt subtraction induces negative numbers. But wait. This is theoretical math, we need to be rigorous. If subtraction is an operation, an operation is supposed to be defined ... and an operation is defined on a set: it takes two elements of the set and gives out another element of the same set. Now we want to use subtraction, an operation to define integers, but then the definition of the operation needs the the set in the first place ! that is a logical difficulty ... ok, we can pass through this rather quickly ... something like the approach taken by Terrace Tao in his analysis book ... define new numbers a___b [no that is not a-b yet] ... :) we do not understand what these new numbers are , we just define them on natural numbers a and b. some new number for all natural number ... that appears like cheating , but then its completely rigorous ! and then we need to define “equality” - that is important. one can spend a lot of time understanding this equality thing. Different branches of mathematics have different notions of equality -- for example in geometry two objects are equal if i can place one over the other and they match exactly. In topology two objects are equal if one of them can be transformed in "continuous" way into the other !

So we have defined new objects , created notion of equality and we have integers --- now define subtraction, and LO! a___b is just a-b ! :D

3. Rational Numbers:

One might say “enough of integers”, let me journey into rationals and this is easy ... we can mimic the approach we took for integers and then define new objects a//b ... and the rest of the story is similar to integers. Ah , that sounds comfortable – but then we need to see why there can be any number of rationals between rationals and also why there are “gaps” in rationals. Ah, Rationals are incomplete since we have a gap at sqrt(2) ! But why is a “gap” bad ? For all calculations gap is fine – infact the rational number set is very close to integers for all practical purposes. Take a scale, we have 1 cm , and then a 1/2 cm. How about marking it slightly differently ? Lets say i call the 1/2 cm mark 1cm , now the 1cm mark has become 2cm ! So simple rescaling worked here. If we have a finite collection of rationals they can be treated simply as integers ... infact a finite collection of ratioanls can be treated as natural numbers because the computer actually stores everything as binary (natural ) numbers. Ok , but then there are infinite number of these [what does infinite mean ? One can get philosophic about this ... its just something that is not finite] ... and yet they are coutnable ! Thats a new and precise notion... countability ... again that can interest someone and for the next few months or years one may delve on that.

4. Real Numbers:

Now lets fill the gaps in rationals! Why Do we need to fill the gaps ? If we do not fill the gaps ... we miss continuity and if continuity is not there , our picture of motion etc is at risk . We have the so called zeno paradoxes that would tell us there is no motion. So we need these numbers. But do we really solve the zeno paradoxes with the notions of continuity ? Because one can explore the issues from physics angle. For now lets move onto the real numbers ... we found a gap in rationals, let us fill it !

Real numbers are defined as certain sequences of rational numbers called cauchy sequences [there are many ways to construct these but lets use this for now]. So (1,1/2,1/3,1/4,1/5....) represents 0 ! But wait, do you see that (2,2/2,2/3,2/4,2/5,...) also represents 0 ! and there are many such sequences ... so 0 is not one sequence, but a collection of sequences – considered equal. We had this issue with rationals also , 1/2 = 2/4 ... when we have an equality we can use the underlying equivalence relation to partition and use the equivalence classes. And now one might be wondering what is product of sequences and then what is exponentiation? Right ! That is the reason e^x is a very profound term.

But one would perhaps also wonder if these real numbers are only a mathematicians fancy creation and why should the physical world behave according to the fancies of mathematicians ? Or are we simply closing a loop : since i am looking at the world from the angle of real numbers and calculus, it appears to be obeying the logic of math and since math logic seems to be working we are using that more and more to understand world --- perhaps we are simply playing an intellectual game where our own toy world is all we are seeing.

Now one can proceed to complex numbers --- actually i like complex numbers more than the real numbers – they represent certain symmetry operations : namely rotations. Complex numbers, quaternions etc is one way to go about from here. But since i have mentioned calculus – let me also mention another interesting type of numbers “Hyper Reals” --- Hyper reals provide a framework for calculus called non-standard analysis ! And hyper reals are defined as sequences of “reals” ! :) so in hyper reals we can have a sequence that represents a number called infinitesimal --- lets call it : ϵ ... and this is less than all real numbers but greater than 0 ! In the world of hyperreals, there are infinite infinitesimals and infinite numbers and then there is an infinite integer that is greater than the infinite reals ! All these can be freely used in calculations to make calculus a kind of algebra with hyper reals ! This infinitesimal calculus [thats the name ] is equivalent to the calculus of standard analysis – that means when we say motion is rate of change of position, we are talking of all these “hyper numbers” in some sense ! Wow !

The story does not end here , there is lot more to write. We just touched upon one area : analysis and basics of number system. There are lot of other topics that are equally [if not more] interesting. But now its time to end this , leaving another discussion for a later time.