Take care and watch this space.

The approach taken in this blog for developing the physical theories will be the axiomatic one. I’ll do this because of brevity, internal elegance and consistency. Of course, I’m well aware of the fact that this is only possible with hindsight but I think that one has a lot to gain when physics is presented this way. Maybe the one who has more to gain is the presenter than the presentee, but since this is my blog I’m calling all the shots.

Maybe a word is in order for what the word axiom means and a little bit of history will be needed (gasp!!! the first self-contradiction!!!). In ancient Greece, the place where normally one thinks real science started to take shape (actually it wasn’t but this is a whole other can of worms), people who concerned themselves with such matters used two words to signify two things that nowadays are taken as synonyms. Those two words were: axiom and postulate.

Back in the day axiom was taken to be a self-evident truth while a postulate was taken to be something that one would have to take for certain for the sake of constructing an argument. So, axiom was a deep truth of nature while a postulate was something that humans had to resort to in order to reach new knowledge.

As an example of an axiom we have Euclid’s fifth (which revealed itself to be quite the deep mathematical truth!) and as an example of a postulate one has the assumption that Hipparchus made that the sun rays travelled in straight lines from the Sun to the Earth and Moon while he calculated the distances and sizes of those three bodies.

People have become a lot more cynical and in modern day usage those two terms are used as synonyms (and the meaning that prevails is the postulate one).

Axioms arise in Mathematics when one is willing to construct a theory that will unify a body of (not so) disjoint facts into a coherent whole. One should take proper care that the propositions one uses as the building blocks are enough for completeness and internal coherence and can derive the maximum amount of new facts with the minimum amount of assumed propositions.

In Physics things seem to be different at first sight but let me show you that things aren’t that different after all. For starters one knows ever since Galileo that the verbal method of Aristotle – (metha)physics – isn’t the way to go for one to know, predict and even interfere in natural phenomena. For all of this to happen mathematical tools are needed. One gets deeper into the truth of things, and one is also able to get technological progress that, besides of messing up with the natural environment, also makes people’s life easier. It isn’t enough to tell that bodies fall under gravity, one has also to specify where, with what energy, under what time interval such a fall happens.

For instance Newton’s theory as it was done by Newton was axiomatic. His three laws are just another name for axioms. They are the propositions that contain the undefined terms whose validity one has to accept in order to achieve new results.

One fundamental difference now arises. While in Mathematics things are normally evaluated in terms of self-consistency and internal elegance (this is a HUGE oversimplification) in physics things are also judged by how good the new results compare to actual measurements in the real world. In Physics physical theories have to be consistent with what see around us (another HUGE oversimplification). Hence if Newton’s Principia predicted squared orbits for the planets, Newton’s Principia would have to be scrapped.

Another difference is at the way we physicists arrive at the axioms: normally one has some experimental facts and start thinking about them and how they are linked with each other. Hopefully one will then be able to put the most fundamental properties as building blocks of our theories and call them axioms (in Physics it is more usual to call them laws).

After digressing a little, thanks for reading by the way, let me proceed with the defense of the axiomatic way in Physics. One other thing is that I think that knowledge is a lot more sound when one knows where one stands and why one is standing there and not some other place. So, if  I tell you what are our basics (it doesn’t matter how we get to them) and derive all that can be derived from them I believe that sounder knowledge is achieved.

The historical/phenomenological method has as its big advantage (according to me at least) of showing the inner struggles each concept has to endure before being accepted and being part of the reigning paradigm. It also makes things more approachable at a first attempt, but I think that the merits of this approach stop at this initial pedagogy.

The downsides of the axiomatic way are that, at first sight, it seems highly artificial, and may not be what most people are used to and want to see when wanting to learn physics.

Moving on from this rather big lecture let me explain what I’ll do in the Newtonian Physics part of this blog:

1. I’ll start off by introducing units of measurement, dimensional analysis and explain why they are important in Physics.
2. A little bit on error propagation and why it matters in physics. Yes, this is mostly a theoretical blog but I consider this to be part of the physicist knowing where he/she stands paradigm.
3. Assume that the reader knows differential and integral calculus (even though I’ll continue my posts on Basic Mathematics).
4. Introduce the Newtonian axioms and what most people think Newton meant to say what while introducing them.
5. Do a lot of calculations.
6. Have a lot of fun!

I still intend to follow the plan I laid out in the beginning and I understand if some people are getting turned of by the long bouts of inactivity this blog has (the next post is almost finished though).

For the ones of you that just can’t wait to get your Physics right away you can go to this other blog of mine: ateixeira. I’m starting with the Physics right away (and I think that in the future some of what I’ll write there will end up in here), even though it’ll be on a somewhat advanced level.

So, if you just can’t wait to get to Physics just click here and get done with it!

O objectivo do blog é produzir nova Física através da interacção dos seus membros e ser também uma plataforma de ensino e discussão da Física (e áreas directamente relacionadas) em português sem ter medo de se recorrer a equações.

Visitem, comentem e divulguem.

— for Continuity —

First we’ll start things off with something really simple.

Let which is obviously continuous.

The gist of the the reasoning is that we want to show that no matter the that is chosen at first it is always possible to find an that satisfies Heine’s criterion for continuity.

Getting back to our function it is . Here so

Which is trivially true since by assumption. Hence any value of will satisfy Heine’s criterion for continuity and is continuous at .

Since we never made any assumption about other than we conclude that is continuous in all points of its domain.

Let us now look at . Again we’ll look at continuity for point ():

The last expression is just we want at this stage since want to have something of the form (the first part of the criterion).

If we let it is and this completes our proof that is continuous at point .

And again since we never made any assumption about other than we conclude that is continuous in all points of its domain.

Now we let and will see if is continuous at .

Hence if we let it is and is continuous at .

As a final example of Heine’s criterion of continuity we’ll look into .

Since we want something like the last expression isn’t very useful to us.

In this case we’ll take an alternative approach which nevertheless works and has exactly the same spirit of what we’ve using so far.

Please look at every step I make with a critical eye and see if you can really understand what’s going on with this deduction.

Since we know that at some point will be in the first quadrant. Thus

Where the last inequality follows by hypothesis.

That is to say that if we let it is which is the epsilon delta definition of continuity.

— for Limits —

After looking into some simple proofs for continuity we’ll take a look at for limits.

The procedure is the same, but we’ll state it explicitly so that people can see it in action.

Let . We want to show that it is .

Which is trivially true for any value of , hence can be any positive real number.

Let . We want to show that it is .

With we satisfy the for limit.

As a final example let us look at the modified Dirichlet function that was introduced at this post.

At that post it was proved that for didn’t exist and it was promised that in a later date I’d show that using the epsilon delta condition.

Since we now know what the epsilon delta condition is and already have some experience with it will tackle this somewhat more abstruse problem.

Since or we have two cases to look at.

In the first case it is which is trivially valid, hence can be any real positive number.

In the second case it is . Hence letting gets the job done.

Since we proved that the conclusion is that the modified Dirichlet function that was presented is only continuous at .

As was said previously, they don’t make local concepts more local than that.