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!