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There is a lot here

1. Do mathmaticians sometimes disagree?

Of course they do. One might think that vanilla ice cream is better than chocolate ice cream and another the opposite. We can disagree about a lot of things. Granted, what ice cream is best doesn't really relate much to mathematics. Another example is that mathematicians (as teachers) can also disagree about how to teach mathematics. While the questions about how to best teach mathematics might not seem to relate to the fundamental nature of mathematics, there is a relation.

Mathematicians will also disagree about philosophy. How do we best think about mathematics? What is the usefulness of mathematics? What philosophical traditions lead to society's understanding of what mathematics is? What is mathematics? All these questions are very interesting to consider, but one might not want to call them actual math questions (I guess people will disagree about that).

But, when you compare mathematicians to other groups of people, I believe that you will find that mathematicians are (in general) in much greater agreement than disagreement about what they actually do. If I present a result at a conference and if I have published an article, then it doesn't happen often that a audience member will say that he disagrees. He might point out that I have made a mistake in my reasoning, but he wouldn't say that he disagrees on some principal grounds (in general!). If a mistake is pointed out to me, then I go home and try to fix it. I don't whip out a long discussion. My focus would be on trying to (1) determine is there indeed is a mistake, (2) correct the possible mistake. But, sure, you might find yourself in a situation where two mathematicians disagree about whether or not there even is a mistake.

Do mathematicians disagree that the derivative of the function $f(x) = x^2$ is $f'(x) =2x$? No, I would be hard to get a job at a research institution if you at the job interview said that you didn't believe that this is true.

So why do mathematicians agree more? I think it has to do with the very nature of mathematics. In mathematics we have definitions that one can't disagree with. One can say that the definition should be different, but mathematicians are in general fine with you making your definitions as long as your stay consistent because they know that you can rewrite your results to adjust for the change in definition. As an example: Is $0$ a natural number? Here people will disagree, but it doesn't matter. It often comes down to making your theorems easier to state. If I, for example, have to talk about the set $\{0,1,2,\dots\}$ a lot, I might just first define the symbol $\mathbb{N}$ to mean this set. Otherwise I would have to invent new (and more complicated notation like $\mathbb{N}_0$ for such a set and I might not want to do that (I think you should by the way).

2. About the video.

So I actually watched the firsst 17 minutes and 34 seconds of the video. This is the part where N J Wildberger (?) talks about why he doesn't "believe" in infinity. He tries to present an argument that infinity doesn't exist.

Some key points of his argument are

Since something is new, it probably isn't true. He quotes several philosophers and says that Cantor would be surprised if he knew where we had taken mathematics. My response is: Ok, so? Mathematics isn't a game of how to historically back up our arguments. Even though the Egyptians were successful in applying there method of doing arithmetic doesn't prove that it is superior. If you take a class on theology (say), then indeed some will say that a certain view point is false because it contradicts the way we have always done things. But, hopefully, you can see that this style of reasoning doesn't go well in mathematics. Also, historically mathematicians disagreed more maybe because of the closer connection with philosophy. Many mathematicians were philosophers. What has been attempted (and to some extend accomplished) in out day is a greater formalization of mathematics. This formalization removes much of the disagreements. Since the computer scientists would disagree it must be false. It is also hilarious how he keeps making references to computer scientists. The reason this is so funny is because: computer scientists are not (in general) mathematicians! Just because a computer deals with finite "things", doesn't mean that mathematics can't deal with infinite "things". Listen to the first 17 min and 34 sec and count the number of times this is his primary argument! Just because it is hard to construct much mean that it doesn't exist. This is a philosophical argument. If he had said: Since it is hard/impossible to construct it doesn't have a value for society, it would be easier to agree with him (I would still disagree). This is related to the computers. He writes down a huge number and says that this this number is bigger than the number of anything else in the universe it somehow means that we should talk about infinite "things". Ok, so the number you wrote down is bigger than the number of atoms in the universe, so what? I simply don't see the problem. But I see where he is coming from. He only thinks that mathematics is what can be constructed. This is strange since a mathematician would know that proving existence of something is often very different from constructing it. Because my calculus book doesn't define (construct?) the real numbers, they don't exist. Ok, I am not being nice here. But don't you think it is funny that he quotes a calculus book when talking about the definition of the real numbers? Sure, his point does show that we, in calculus, deal with sets without having defined them. We work with the real numbers without actually constructing/defining them from the axioms of set theory. But why is that a problem? I agree that on some intellectual level this is a problem. But in terms of teaching students how to take derivatives, maybe it is ok? (Again a teaching question that we can disagree on!)

In the end, he really doesn't present a very convincing argument.

As he briefly touches upon in the beginning, this all comes down to the axioms. If you take the ZFC axioms of set theory, then there even is an axiom of infinity. So he must be saying/believing that the axioms are not true. But disagreeing with axioms are like disagreeing with definitions!