Foundations of Mathematics II

Okay...the last post needs some clarification. I did not mean the notation '3' for the number '2+1' is unjustified. That is perfectly okay. We can have the symbol '4' for the number '3' or anything for that matter and the math would just be fine and consistent as it is now. So, denoting '3' for '2+1' is understandable and it is not the issue. The point was that how one can assume the existence of a set that satisfies the Peano's axioms? The set that satisfies the conditions of Peano's axioms could well be empty, isn't it? Thus natural numbers are not captured by those axioms rather the axioms hold for what we all know as natural numbers. That is we have to agree on such a basic set in order to move forward.

This points us to the fundamental question on the existence of numbers as basic as the natural numbers. Do they exist at all?

In my view, natural numbers are mathematical representation of the notion of duality that we all perceive, if not take it as a basic philosophical position. The mere fact that we see more than one thing is enough to justify the existence of numbers. In other words, numbers exist and they are real as much as you believe in reading this piece of writing.

Foundations of Mathematics I

Everybody knows what are natural numbers. Or is it so? If I ask, is 2+1 a natural number, everybody (including me) would rush to say it is so - the reason being 3 is a natural number. I agree that 3 is a natural number, but why 2+1? Isn't it silly to question 2+1 = 3? If not, what is 2+1 after all?

The Italian mathematician Giuseppe Peano knew in 1900's that there would be such silly minds like me who will want to question even basic (and trivial) things. So he introduced a set of axioms that would define precisely what those numbers are, in an attempt to capture the notion of a natural number. Here are the axioms:

1. There is a natural number 0.
2. Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
3. There is no natural number whose successor is 0.
4. Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
5. If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers (This postulate ensures that the proof technique of mathematical induction is valid).

It will be very easy to see that the natural numbers satisfy these axioms. But do they define natural numbers or capture the idea of what natural numbers are? The answer is clearly no. I still haven't got the answer why 2+1 should be 3. None of these axioms say it is so. To take a closer look why it is so, consider an alien coming to the earth who don't have the number 3 in their world. He doesn't know what 3 is and in their system 2+1 equals 4. All they have is the set {1, 2, 4, 5,...}. Now, how does one explain our concept of natural numbers to him. Note that these axioms would still be valid in their system.

So Peano hasn't convinced me yet! If one says that there could be some other set of axioms that would precisely define what natural numbers are - then go please find that and let me know. I will give you million dollars, okay 10 dollars. All one can say is that any other set that satisfies these axioms will be isomorphic to the natural numbers. In other words, the word isomorphism here means, I don't know what natural numbers are, but can we all agree there is one so that if you have a different system of numbers then either agree to our agreed definition or go to hell!

Looks like natural numbers are not so natural at all! More on this later.

Why Mathematics Fails?

Albert Einstein said "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". Let us examine why it is so. Consider these two statements (or instances of propositional symbols):

P: I wake up late
Q: I go to office late

The deductive apparatus in formal logic consists of two things: A set of premises (called axioms) and the rules of inference. The rules of inference could be anything like Modus Ponens (MP) or Modus Tollens. MP basically says, the statement if P then Q taken together with P will imply Q. Which kinda looks true in our case: If I wake up late, then I go to office late and I wake up late would mean I go to office late. Among the basic ideas in formal logic is the idea that the deductive apparatus or syntactics should be kept aside from the semantics of the language. One may then naturally ask what is the relation between the syntactics and the semantics of the language. It is here the idea of axioms come in. In general, in mathematics, the main goal is to come up with the set of premises so that the syntactics is matched with the semantics. In other words, we keep the rule of inference fixed and come up with our axioms that deduces statements that are true or false. Although we can change the rules of inference as much as the axioms, mathematicians don't attempt that. If P then Q and P need not always make Q true. For instance, take

P: We men going to mars
Q: Saturn is full of chocolates

Now, if P then Q and P is true does not make Q true - for sure we know that right, although P is a real possibility (and not true now). Mathematicians ignore the general notions of truth and reduce them to validate their statements by building axioms rather than equally exploring the rules of inference. Thus the notion of truth or falsity in mathematics may not conform to the real world. Here is where temporal logic and relevance logic come to rescue!

Immortality Pill.

During our usual lunch conversation, we came across a topic in which we were talking whether the medical science is in the right direction. I was in the position that even if it is in the right direction it is not in the right interests. My friend asked me what if the medical science can discover an immortality pill which can keep us physically alive eternally. The topic shifted direction when I told him that it is possible to be alive for a long time using some yogic methods - with total control of prana. And then I told him that there are yogis who live for more than 200 years in Himalayas and the right approach for the medial science to understand immortality is not through fragmentary and divisive studies but through a unified understanding of what consciousness means.

On retrospect, I think that we don't need to go too far on the idea of immortality and the magic pill the medical field can possibly discover. Even if the medical community discovers a pill that could make us immortal, how does it help from the social conditioning that we find ourselves today? I can still shoot a person at will, can't I?

Vernon Howard

Vernon Howard - It was only last week that I came across this wonderful person. He is one of the enlightened masters and his philosophy looks similar to Jiddu Krishnamurthy's. Here are some YouTube links in which he speaks.

Right Ideas for a Right Day

Way to True Command (3 part series)

Inner Power for Today (3 part series)

Sense of self.

Ultimately everything depends on from where we derive our sense of self. It could be from our culture, religion, political affiliation, nationality, our dress, new hair style and all things external to us - the world of forms. Or it could be from deep within, from our very existence or the feeling of I. In this case, our sense of self is not actually derived from an external entity but it is the feeling of being or completely present. Even the thought can be considered as something external since its away from the feeling of being. Ego is nothing but the derivation of the sense of self from anything away from the complete presence. Being in thought is being in ego. There is nothing wrong to be in ego (we will be forced to) but we will just miss the wonderful feeling of being present, of being alive.

Big Bang in Your Closet?

For long I knew that all the problems of the mind cannot be solved in the level of the mind. For instance, the question of death. Mind as I has envision is like a field which is not algebraically closed just like how the real numbers are insufficient to solve the equation x*x + 1 = 0. However, I did think that the mind is capable of solving almost all analytical questions. By analytical questions I mean the ones that are put forth by science. But yesterday, I had this incredible revelation about a question in science that the mind won't be able to arrive at a solution.

Modern science tells us that there was Big Bang from which the visible universe has expanded from a primordial hot and dense singularity at some finite time in the past. Now we can ask a legitimate question within the domain of science on where was that singularity present - I mean the spatial coordinates of the dense particle. Once we pose this, we realize that the question depends inherently on the notion of distance. Now the concept of distance is a relative measure - we need at least two objects have a distance measure. When all the objects are condensed as a single small package, where does the concept of distance come in? In other words, the Big Bang singularity can never be traced to a specific location in space. Assuming such a thing happened, all we can say is that the singularity could have been anywhere in space. Who knows, it could have been in your closet too.

The Eternal Warrior

Charma Slokam.

sarva-dharman parityajya
mam ekam saranam vraja
aham tvam sarva-papebhyo
moksayisyami ma sucah

Yesterday I was in a lecture by a ISKCON Maharaj and he spoke about this famous Charma Slokam in the Gita (since I requested). I found that the interpretation is different from the one in the Sri Sampradayam. In the ISKCON phylosophy, the word dharman refers to duties of all kinds including any religious obligations one may have. However, in Sri Sampradayam, the word dharman (in Pravritti maargam) refers to all prayaschitta-dharmas, or all obligations taken to compensate for our wrong doings, but the essential dharmas should not be renounced. Bagu (short for Bagavaan) takes care of all the prayaschittas, thus paving the way for bhakti to emerge. In light of Eckhart Tolle, the Sri Sampradhayam interpretation seems right, at least when following the Pravritti maargam.

On Growing Old.

I think I heard this idea somewhere as almost everything for me now looks familiar. Anyway, I will come to the point. The reason for our physical growth is because of the flexibility of our body to cope up with the mind's energy. As soon as our bodies become stiff and inflexible, aging starts, as our mind still works with the same energy. Probably that's why yoga-aasanas helps in slowing the aging process!

Sunset House.

My new painting - (paint by numbers, of course). The original doesn't have the sun like bright spot in the middle - that's the flash glare. It looked nice and I left it alone.