The answer is "who cares?!"

Many things in this world are irrelevant.

Like for example I don't need to know the science behind how a microwave works in order to cook food, or I don't need to know how the internal combustion engine works to drive a car.

Or other silly questions like the question of multiple universes, questions that no-one will ever know the answers to.

Just a pointless use of brain power.

The most important thing you need to learn about maths is that 2+2 is ALWAYS going to equal 4.
It's never going to change, and don't let anyone ever try to teach you different.

Armchair Critic: Mathematicians, Logicians and Philosophers care. But not only them, I dare say that almost everyone would care in certain circumstances. For example, if someone proved tomorrow that 1+0=0 then all employers could refuse to pay their employees because by using this sum it is trivial to prove that any number equals zero, ie: if they owe $1000 dollars to an employee then by manipulating this sum they can show that they really owe $0.

Now this may sound stupid to you, to think that mathematicians could prove a sum like this, but this has actually happened. A bit more than a hundred years ago Bertrand Russell discovered a flaw in mathematics. It was a very, very low level flaw, at the foundational level of set theory. (For most mathematicians, set theory is the lowest level of maths and all results of maths are constructed from set theory. Numbers and operators like addition are actually sets, but very,very complicated ones). Now, any flaw in math, no matter how small, allows you to prove results like the above. See: Russell's paradox on wikipedia for more info. This flaw shocked the mathematical world at the time and people straight away searched for ways to get rid of it. They achieved this by changing the axioms of set theory, that is, they changed at a foundational level what we consider to be true and not true. By doing this, they laid bare for all the world to see that the base axioms and rules of math are arbitrary, you can change them as you see fit. That is, maths can viewed as just a belief system in which you choose what you believe.

Hey thinkabit,

I did pretty well in maths at high school and was placed in an advanced maths class, but I played up in class and they put me back into the normal class because I wasn't applying myself.
I didn't engage in any formal studies after high school Grade 10, so trying to understand Russel's Paradox and this 'set theory' stuff is a little above my pay grade.
- I don't understand what the strange looking symbols mean.

For me though, things don't need to be so complicated like I said earlier.
Take your example- 1 + 0
Lets say 1 = Something;
and 0 = Nothing

You cant start with something, add nothing, and have a total of nothing, its impossible.

2+2 will ALWAYS equal 4.
To entertain the idea that adding 2 and 2 can result in any number other than 4 is to me absurd, like 'you are now entering the twilight zone' absurd.
(So you we're right - it does sound stupid to me)

But I must say I wasn't aware of Russel's paradox before you mentioned it and I'm glad you did, as I found this trivial little bit of learning curiously interesting.
Now I understand where people get this 2+2=5 stuff from.

For me though there are no flaws in maths.
I like the term 'mathematical certainty'.
So I'm thinking there must be a flaw in this 'set theory' instead.
I don't think maths should be considered a belief system in which you choose what you believe.
I don't even know why they would try to teach people that, but hey, I'm not a Mathematician, Logician or Philosopher.

Thanks for the info.

Kurt Gödel's incompleteness theorems established that there is uncertainty in mathematics.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Hello davidf,
I was going to mention Godel's theorems in my post above, but decided that it would complicate the post. It does fill out/complete the story of people searching for a way to remove Russell's paradox from maths.

Hello ArmchairCritic,
You seem absolutely certain that you "know" there are no flaws in maths. The question I have for you is: How do you know?
If you say something like "Because I just know that it is true" then you've basically admittedly that it is something you believe purely by blind faith. Believing something is this way is no different from a Muslim saying that he knows that the Koran is the inspired word of god that we must submit to its teachings or a Christian saying that she believes that "Jesus is the way, the truth and the light and no man can enter heaven except by him". Ie, for you maths is a belief system taken on blind faith.

If on the other hand, you may say something like "Maths is true because whenever you look at the world around you, when you add 1+1 you always get 2, eg: one apple + one apple always gives you two apples.". In this case, you are saying that maths is a way of summarizing the results of empirical observation of the universe. That is, for you maths is a science and we find out about math by systematic experiment on the "real world".

Choosing one of these two views points as the proper view point is what the original post of this thread is about.

However, these two views don't encompass all current schools of thought on maths, there are other ways which you can answer this question. Read the Wikipeadia article on the philosophy of maths for more.

Dear Armchair Critic,

Not everyone is happy to go through life being an ignorant person. There are a lot of us who want to know about the world and universe we live in; and people like me who also want to know everything about humankind.

That is why we study the Sciences and Arts so that we don't have to go through life being ignorant. For people like you ignorance is bliss but for a lot of others ignorance is anathema to their very being.