Someone told me that math has a lot of contradictions.
Correct mathematics does not, as far as we know. However, mathematicians amuse themselves with little "proofs" whose conclusion is absurd. The game is to identify the error. It's important because the "proofs" usually rely on errors that people often make by accident. Finding the error helps mathematicians avoid making the same error themselves.
Probably the simplest such game is a "proof" that $1 = 0$:
Let $x = 1$ and $y = 0$. Then $x \cdot y = y \cdot y$. Dividing both sides by $y$, $x = y$.
The error of course is that you cannot divide both sides by zero and maintain an equality. The lesson is not to divide by something that is or might be $0$. In a more complex proof it might take some work to prove that the thing you want to divide by actually is never $0$, or if it could be $0$ to consider the case where it is separately from the case where it isn't.
Sometimes this game becomes more serious. There was a thing that is now called "naive set theory" that basically said, "any collection of sets that you can describe, is a set". This allows us to consider such things as "the set of all sets", and "the set of all sets that are elements of themselves". Of course, "the set of all sets" is an element of itself. The empty set is not an element of itself (no set is an element of the empty set, that's what empty means). So, what if I define S to be "the set of all sets that are not elements of themselves".
Oh dear. Now we have a contradiction. Is S an element of S or not? If it is, then it must satisfy the definition. But according to the definition anything in S is not an element of itself. And "itself" is S. That it to say, if it is an element then it isn't. And if it isn't an element of S, then by the definition it is an element of itself. So if it isn't an element then it is.
What this told mathematicians is that what is now called "naive set theory", just defining any old thing you like, isn't good enough. This is called Russell's Paradox, because Bertrand Russell published it first. Ernst Zermelo had previously discovered it but not published. Both Russell and Zermelo set about constructing systems that:
- allowed everything that mathematicians needed to do with sets (actually Russell worked on "type theory" rather than "set theory", but I don't think the difference matters to this explanation).
- prevented the paradox from occurring by limiting how you're allowed to define sets.
You could reasonably argue that prior to this activity, mathematics did contain contradictions. Fortunately not in a way that mattered, since as it happened there weren't any really important results that couldn't be brought onto more secure foundations. But for this and other reasons we cannot say for sure there are no contradictions, only that there are none we know about and haven't dealt with.
This might sound like a terrible crisis in mathematics, finding "math" to be contradictory. In some sense it was a crisis, in that it required a lot of re-checking of some basic assumptions and intuitions people had. It wasn't a disaster, since set theory as such had existed for less than 50 years at the time, and all the paradox said was that set theory wasn't quite right and needed improvement. Most of all mathematics ever done at the time had been without any particular appeal to this flawed set theory.
You can also think of it as an extended and surprising "proof by contradiction". A proof by contradiction says:
- suppose X is true
- deduce a contradiction
- conclude that X is not true
Set theoreticians had:
- constructed a theory of sets
- deduced a contradiction
- concluded that this theory of sets was no good
So mathematics does "involve" a lot of contradictions in the sense that a lot of proofs you see will end with one. Doesn't quite mean that it "contains" them :-)
He said that a lot of things are not well defined.
This is arguable at a stretch. In mathematics pretty much everything is defined, but one interesting question is what it's defined in terms of. Foundations of mathematics is a large subject within mathematics.
Not true, but you can make it look true in a variety of ways. This is one of those amusements I mentioned above.
Given a divergent series, you can perform some incorrect but plausible-looking manipulations that result in it appearing to have any total you please. AFAIK the original historical reason for choosing
-1/12 in particular, is that
1+2+3+4+... is the divergent Dirichlet series for
-1/12 is the Riemann zeta function of
-1. Now, when the Dirichlet series converges for a particular value, it is equal to the zeta function. In fact the zeta function can be defined as the "analytic continuation" of those convergent Dirichlet series. That means that it's the only function that's equal to the convergent Dirichlet series and also has another special property called "holomorphism".
But the Dirichlet series for
-1 doesn't converge, as I said to begin with it's divergent. So
-1/12 is the value of an important and interesting function that coincides with some Dirichlet series, but not this one, which doesn't have a value. So you can call it "the value of that Dirichlet series", but really it isn't. There is no contradiction.
what is infinity ∞?
Since you're not mathematically trained you can't be expected to know what infinity "is" mathematically. Quite aside from the fact that the concept of "infinity" is somewhat mysterious in English, your friend has asked you an unfair question of mathematical general knowledge. He might as well have asked you what the Riemann zeta function is, or the definition of a metric space, and if you've never studied those things then you just don't know.
In mathematics you deal with the concept of infinity by defining very strict rules for how to handle it, and then following them. In different contexts mathematicians will define "infinity" differently. So you should think of ∞ as just being a symbol used to mean some specific thing that's defined somewhere else (hopefully the person who uses it can tell you where). It doesn't necessarily mean the same thing every time it's used in different places.
Back in that subject, "foundations of mathematics", there's plenty of interesting work on how to consider the infinity of the natural numbers 0, 1, 2, 3, ... Further, there's interesting work on whether mathematicians "should" be considering infinity as a quantity at all. Since it's foundational this work can in principle affect all of mathematics, in practice most areas of mathematics can just pick the version of "infinity" they need (if any) and stick with it.
How to disprove the previous two?
I can think of several possibilities:
your friend has genuine questions about the foundation of mathematics. This is fine, but you, as a non-math-specialist, are not going to deal with his concerns. Refer him to a mathematician who will make him be more clear exactly what his concerns are: what does he feel is not well defined? How he has derived his contradictions?
your friend has encountered something he doesn't fully understand, and concluded that it is ill-defined and contradictory. That's a natural response, but in the case of mathematics there's an antidote, which is to ask genuine questions about the foundations of mathematics. See above.
your friend is messing with you. Since he has offered you no proof of what he says, there is no need to find a particular error. It's as if he said, "English contains a lot of contradictions. For example the word 'red' means 'orange'". Well firstly, no it doesn't, it means something different, you can just ignore him (and the sum of 1+2+3+4+... is not -1/12, that series doesn't have a sum). But secondly, OK, in certain contexts actually it does mean that for interesting historical reasons. In fact the word "orange" entered the English language long after certain orange things had already been named "red", such as redheads, red kites, and robins' red breasts. But this is not a contradiction (and in the context of Dirichlet series and the Riemann zeta function, -1/12 is in a strange sense what the sum of 1+2+3+4+... "would be if it converged").