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I know it's a off beat question but I thought 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 convinces 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 how 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 you for example 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 if in physics, you have great minds like Albert Einstein, who was convinced that "God doesn't play dice." and disagreeing about this subject with other scientist until the end of his life.

So to my experience, this doesn't apply to mathematics so much. The only nowadays mathematician that I've ever heard of that strongly disagrees with other mathematician 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 are my question:

Am I right, that almost all mathematician, in the end, agree about things in mathematics ? Or are there much 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 my own ideas about this, but I would like to hear others about this. What is the big difference between mathematics and 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 ?

  • The basic "disagreement" between mathematicians communities regards some fundamental issues : see Constructive Mathematics and Intuitionistic Logic and Intuitionism agains "mainstream" or classical mathematics. – Mauro ALLEGRANZA Oct 29 '14 at 12:30
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    Math is generally NOT considered a science, because, e.g., it can't be disproven experimentally -- a difference which helps to explain its greater certainties. It is of a fundamentally different nature than science. – selfConceivedAsEvil Oct 29 '14 at 13:06
  • ...put another way, I think there are mathematicians that see math as a science, but they disagree with most other mathematicians in this sense ;) – selfConceivedAsEvil Oct 29 '14 at 13:21
  • @goldilocks - are you sure ??? See University of Cambridge : Colleges and departments : it happens that you can find the Faculty of Mathematics is the School/Dept of Physical Sciences and not in that of Arts and Humanities... – Mauro ALLEGRANZA Oct 29 '14 at 13:30
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    Mathematicians agree at the end because they agree at the beginning. – Hunan Rostomyan Oct 29 '14 at 14:56
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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.

  • I wouldn't really go so far to call it an 'art' either, but agreed it is definitely not a science. – surelyourejoking Oct 31 '14 at 13:14
  • A form of mathematics as arbitrary as your description of modern math indicates would surely lose the respect of the culture as a whole. Math may not study objective reality, but if it turned up its nose on real concerns this much, it would fail to be an important fixture in our academic system. – jobermark Oct 31 '14 at 19:08
  • @ Jobermark But that's how pure math is, now. It so happens that many results from pure math find applications. However, lines are blurred between pure and applied math in several sub-disciplines, so yes, it's debatable, I suppose. – Aditya Kashi Nov 1 '14 at 22:08
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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.

0

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

  • First off welcome to philosophy.se. I like that what you've written is both concise and gives at least one example. But thinking about it, I'm not a little lost as to what it means. Can you give some sort of criterion for what makes a question "advanced" or "simple"? – virmaior Oct 30 '14 at 10:49
  • I would claim quibbling over definitions is not genuine disagreement, but just that -- quibbling. – jobermark Oct 30 '14 at 22:20

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