Do mathematicians always agree at the end?

Question

I know it's an off beat question but I thought a philosophical answer would be better. I've been trying to study some different sciences in my life, ranging from biology to mathematics, and if I try to explain to people why I like mathematics above the others, I think the most important reason for me is that mathematicians, in the end, almost always seem agree about something.

I mean, sure, sometimes, I disagree about something with some other student, but I'm sure that either he can convince me that I am wrong, or I can convince him that he is wrong. Or if we are really stubborn, I'm sure that we can go to a teacher, and however stubborn we may be, in the end, one will be convinced that he is actually wrong.

Well, in all other sciences, the opposite seems to be true. If for example you look at health sciences, then you hear a scientist, that studied this matter for years say almost the exact opposite of some other scientist. And those scientists debate with each other, and in the end they still disagree.

Even in physics, you have great minds like Albert Einstein, who was convinced that "God doesn't play dice," disagreeing about this subject with other scientists until the end of his life.

So in my experience, this doesn't apply to mathematics so much. The only mathematician nowadays that I've ever heard of that strongly disagrees with other mathematicians is N J Wildberger. I was was watching this video,

https://www.youtube.com/watch?v=5CiiGdaYEPU

where he is trying to convince the audience why they should change their mathematical point of view. What interested me most is that he claims that historically mathematicians disagreed much more than we do now, which I wasn't really aware of.

So here is my question:

Am I right, that almost all mathematicians, in the end, agree about things in mathematics? Or are there many more mathematicians like NJ Wildberger that I'm not aware of?

If I'm right in (1), I'm curious, what makes mathematics so that mathematicians agree? I've got my own ideas about this, but I would like to hear others about this. What is the big difference between mathematics and the other sciences that makes mathematicians agree much more. And if I'm wrong in (1), can you give me some nowadays mathematical debates, where those disagreements are discussed.

Is NJ Wildberger right that in the past mathematicians disagreed much more?

Answer 1

Usually, pure mathematics is considered an art rather than a science. It rarely deals with "reality"; it hardly concerns itself with observation of the universe.

So if a mathematician invents a mathematical structure, and proves certain statements about his structure using rules of logic, and shares them with other mathematicians, there's hardly anything to disagree about.

This is true about modern axiomatic mathematics. But earlier, mathematics was more concerned with reality than pure math is now. In many cases, mathematics was inseparably mixed with physics, economics, astronomy etc. So it's probably true that mathematicians disagreed more in the past.

I'm sure there are disagreements in applied mathematics, though, since it's about finding, and working with, mathematical models that describe some phenomenon. But applied mathematicians (or people working in the 'mathematical sciences') probably disagree less compared to people working in other sciences because a numerical computation or experiment can usually prove whether a model works or not.

Answer 2

I would argue from a partially Intuitionist point of view that mathematics is the study of human idealization, and not of any trans-human aspect of the real world, nor of some specific ideal world. The reason for eventual convergence, then, is that there is a shared human psychology with a strong tendency to generate similar idealizations in different minds. Mathematics is, put short, the oldest branch of psychology.

I agree with Lakatos, that mathematics is a science, and that we should have a model of science that includes mathematics as an instance, or our model of other highly mathematical sciences will necessarily become inappropriate. But it is an odd science, with only a few close siblings.

It is not about objective reality directly, but it is not totally divorced from it either, as it is elements of the outside world that motivate the direction we move in the exploration of mathematical structures. We do not choose them at random. (Or when we do, like in the case of Algebraic K-Theory, it is because we think they share some 'essential beauty' with something that was directly motivated by the outside world.)

This puts mathematics into a class with linguistics and analytic psychology, and to some degree music theory, where we are studying projections of ourselves, which are only partially objective, and are mostly made up of rules.

When we back off a few steps outside the shared ideal world, or when we try to push it into areas where the intuition is not shared, or where the nature of language fails us, mathematicians will not ultimately agree. This is most obvious in foundations, where we know the set of deep, basic principles most natural to humans is inconsistent. (Russell, yada, yada, yada.) But outside of these philosophical fringes, we expect convergence. This is an assumption the intuition makes about itself on the basis of its intention to facilitate communication, and not an observation.

We do see similar convergence in linguistic structures, so deep that children seldom make grammatical errors that would not be correct formations in some existing grammar, and in music, where dissonance and complexity seem to be real, objective aspects shared between humans in general -- despite the relatively arbitrary distinctions between levels of detail that all simply describe combinations of frequencies. But we should not presume this is perfect, and that in the fringes we will eventually decide upon a correct interpretation.

Answer 3

No. There are schools of thought, with many a disagreement about fundamental and philosophical / metaphysical questions regarding what maths is about etc.

However, they do tend to agree on those theorems that can be proved within two or more schools. You see, mathematicians tend to be smart, and more liable to say something sensible like Hang on, maybe we are disagreeing only because we are not really talking about the same thing.

Answer 4

When considering more advanced questions it seems to be more common for mathematicians to always, in the end, reach the same conclusion.

However, there are some simple questions, such as "is 0 a natural number?" I'll include a link to a discussion on this topic.

https://math.stackexchange.com/questions/283/is-0-a-natural-number