Do mathematicians, in the end, always agree?

Question

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 to 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:

1. 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 ?

2. 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.

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

Answer 1

As an example of a mathematician who disagrees with lots of things in standard mathematics, you might consider Doron Zeilberger. He has lots of controversial opinions, the main one being that demanding rigorous proof is counterproductive (we'd be better off having a more 'experimental' view of truth). He sometimes comes close to saying that mathematics using infinities is wrong (or more accurately, meaningless).

Answer 2

Generally, there is not much disagreement over what has been found. There is a lot of disagreement over where to dig and the importance of what has been found.

Answer 3

The black and white answer is "no." The real answer is more like "no, but..."

Not all mathematicians agree.

As Zubin points out in his comment, many people have debated over the Axiom of Choice. The Axiom of Choice itself seems innocent enough, but it can be used to come up with theorems like the Banach-Tarski theorem, which makes more than a handful of people uncomfortable.

There are also more subtle differences. The current math class I'm taking at my local university is "Hyperbolic Geometry." My professor, a fairly well-known combinatorialist, insists on using an entirely "visual" approach, particularly favoring the use of small triangles in the hyperbolic plane to approximate Euclidean triangles. Being a wannabe geometer, I would prefer to use Riemannian geometry. Though we disagree, our approaches are essentially equivalent. So, in a sense, we can disagree on aesthetics too, but this doesn't often contradict our conclusions.

That being said, mathematicians tend to agree on most things because math is highly rigorous. Opinions don't really count once axioms and logic are aligned. In other words, if it's the truth, then it is the truth.

Answer 4

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

1. 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.

2. 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!

3. 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.

4. 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!

Answer 5

You have the comparison between mathematics and physics backwards, actually. There are several virulent disagreements in mathematics, for example between standard and non-standard analysis or between classical and constructive logic. The latter means that mathematicians cannot in fact even agree on 'what follows from what', since it (can) depend on which logic you believe in. There is a (large) 'mainstream' school of mathematics and smaller schools that disagree with it, but certainly no 'in the end everyone agrees'.

Debates are actually really difficult in mathematics. The first problem is that the assumptions themselves are supposed to be 'a priori plausible', and not everyone agrees on what that means --- the disadvantage of being a deductive science is that it's hard to get consensus on what to deduct from! So for issues like classical vs. constructive logic, the side a mathematician chooses tends to be logically prior to any possible evidence. Furthermore, even when evidence is available, it tends to take the form of 'here is a conclusion you can draw from those assumptions; it justifies / undermines the usefulness of the assumptions'. But deciding which it does still requires mathematicians to decide whether they like / dislike the result, which is subjective in itself.

By contrast, while it's true that Einstein thought quantum mechanics was incomplete (in the 1930s!), not even he thought the quantum mechanical results were wrong, per se; he thought there was a deeper, non-quantum mechanical theory underlying them. And, remember, this was about the time the electron was being solved (WikiQuote traces it to 1926 originally) and before the proton or neutron had been solved. Einstein died in 1955, and the quark was first proposed in 1964. So it's much more fair to say "in the end, physicists all agree" than to say "in the end, mathematicians all agree".

Answer 6

Three points-

$1$ While mathematicians do make mistakes, once the mistakes are spotted, consensus is easily achieved.

$2$ Consensus is not achieved through coercion or because of some social interests; there is a shared genuine conviction that is acquired independently of the opinion of others (the arguments might be displayed by others, but the opinion is formed only in view of the arguments themselves, not in view of who has formulated the arguments).

$3$ Internalisation does play a role, since mathematics is learned, but "homogeneous indoctrination" or "internalisation of social standards" is not an explanation of homogeneous opinion on a proof.

Some philosopher asked once that If we are all rational, provided we have access to the same information, shouldn't we all agree, not only in the science actually, but even in our opinions on all matters? Why don't we?

Descartes had an answer: "The diversity of our opinions arises solely from this, that we conduct our thoughts along different ways, and do not fix our attention on the same objects."

So for him, it is not just the issue of the avalaibility of information but also of the attention we pay to different bits of it. But shouldn't our attention itself be guided by reason or at least by some form of cognitive efficiency? Compare with another cognitive ability, vision. We expect people to agree about what there is in front of them (not necessarily the interpretation of what there is, but some basic identification of items in the visual field) even if their attention is not indentically allocated. Nor so with reasoning. We are used to disagreement in all matters where we arrive at our opinions through arguments. And yes, mathematics is, in this respect, an exception or at least a limiting case of minimal or no disagreement.Could the explanation, or part thereof, be along the Cartesian line: in maths - unlike what happens in other domains -, the same information is available to all mathematicians, and moreover, on mathematical issues, their attention to the relevant mathematical facts and tools easily converges? (and this relates, of course, to the problem raised by Palma of what this information is about, i.e. on the peculiar metaphysics of mathematical objects.)

Answer 7

A partial answer:

In Edward Nelson’s (2007) review of the book 18 Unconventional Essays on the Nature of Mathematics (American Mathematical Monthly, vol. 114, pp. 843–848), he tries to answer the question “Why is it that mathematicians are such nice people?”

We [mathematicians] are no respecters of persons (in that curious phrase that means we do respect persons but pay little attention to the trappings of age, position, or prestige), we take equal delight in fierce competition and collaborative effort, and we are quick to say “I was wrong.” Perhaps some of us know an exception that proves the rule, but by and large I speak sooth, especially when one compares mathematicians to our colleagues in the humanities.

How does one explain that we are so lovable? Is there something in the nature of mathematics that attracts gentle souls? Possibly, but another explanation is more convincing. We are singularly blessed in that the worth of a mathematical work is judged largely by whether the proof is correct, and this is something on which we all agree (eventually), despite the fact that we may have divergent views on the nature of mathematics [...]. This is a singular fact. In art, projection of personality may prevail; in the humanities, the power of position may prevail; in science, the prevailing fad may prevent the publication even of excellent work—but we are extraordinarily fortunate that in our field none of this matters.

Answer 8

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 ?

The real issue here is whether "does infinity exist?" is a question in mathematics. On the video, Wildberger describes it as a question for philosophy of mathematics, which seems like a better classification of the question.

Of course, there is a very large and fruitful relationship between philosophy of mathematics and mathematics itself, particularly in the field of mathematical logic and foundations of mathematics.

One nice contemporary example of this is work by Joel David Hamkins and collaborators on the "multiverse" approach to set theory. But the real "disagreement" there - which is about whether this approach is the right way to do set theory, or whether there is "really" a multiverse of sets - is arguably not a mathematical question. It's unlikely that the two "sides" disagree over any of the mathematical aspects of the argument: they disagree over their personal interpretation of those aspects.

Perhaps this is no different than two chemists who agree on the chemical composition of tea with milk and tea without milk but can't agree on which one is a better beverage. Surely that is not a question about chemistry...

Answer 9

If you consider statistics as well, then there is a large amount of disagreement between frequentists and Bayesians, which has been going on for a while.

Answer 10

Regarding:

in the past, mathematicians disagreed much more?

that is a GREAT question.

I BELIEVE you could consider that to be the case, because, when new concepts came along, notably negatives, and then imaginary numbers, there was a lot of resistance at first - a tremendous disagreement.

As I understand it, there is less of this today, as the very nature of abstracting outwards evermore has been largely agreed upon as a kind of underlying sensibility.

Previously whenever anyone abstracted outwards, there was a lot of resistance.

For an extremely long-winded discussion on this, I direct you to the fairly recent book

Surfaces and Essences:
Analogy As the Fuel and Fire of Thinking
19 mai 2011
de Douglas R. Hofstadter (Auteur),
Emmanuel Sander (Auteur)

Other history-of-math experts here may have more and bigger takes on this.

Answer 11

A famous mathematical result that has been proven by the reknown game theorist Robert J. Aumann states that

"We cannot agree to disagree"

Actually this result admits a surprisingly simple mathematical proof based on assumptions that are indeed fulfilled in everyday life. However, the process of reaching agreement is iterative and might take some time.

So, although mathematicians do not always agree, they have a proof that at the end they will! (Corollary: And if they still don't agree, then that is not the end... )

R. J. Aumann,"Agreeing to Disagree," Annals of Statistics 4 (1976), pp. 1236-1239.

Answer 12

IMO, that depends highly on the realm of mathematics being discussed.

Since they can become highly specialized fields, and the foundation knowledge of one does not always translate directly to the other (basic accepted forms of math aside), the question can become almost as ambiguous as "How is the best way to write this in C++" There can be many opinions, and many theories, some can even provide a modicum of proof to substantiate the claim. But keep in mind some math problems as of yet have no solution, and therefore no definitive reference as to whose theory is better.

Eventually they slip into the arena of who can prove me wrong, then enters the minds of everything from ancient aliens to theoretical physicists...

That leads to an in general answer of no, math is a constant in all forms, but the practice of math in one form or the other in a highly specific field can lead to fierce argument over best approach, most viable theory, and validity of claims based on the simple absence of credible counter argument.

Answer 13

No.

However, they tend to wait till the disagreeing party dies, which makes all the survivors agree. Look for example at Fourier who had Lagrange against him.

Answer 14

My two cents. I think all mathematicians are constrained to abide by the rule of Law (already made) that is rock-like, canonical, thing given to us no one can change , can only add to the foundation. Even if what is now being made through own hands looks so flexible like molten lava in the flow, all unravelings are niches in an evolving world of harmonious music that later on turn to rock for future a reference, a hope that fuels new logic and intuition recipes during the self-expression process. In a fluid ambiance of course, agreeing to disagree is quite apart of the flow.

Answer 15

Mathematicians agree over small matters in the end, yes, but there have been, there still are and there will always be plenty of controversies and disagreements. Usually these disagreements are not over particular results but over the direction that mathematics should take. This article should be relevant.

Examples:

1. Georg Cantor's most influential work was highly controversial before general acceptance. Read the introduction of his wikipedia entry.
2. Grandi's series was the source of dispute until the notion of infinite series was solidified and the series could be classified as divergent (under the standard definition of summation).

Answer 16

"Or are there much more mathematicians like NJ Wildberger that I'm not aware of ?"

There exist more than you're aware of. In addition what others have pointed out here, many mathematicians accept the axiom of extensionality. On the other hand, fuzzy subset theorists reject the axiom of extensionality (at least when doing fuzzy theory). Some people (not all of whom are currently living) who have written on fuzzy set theory are Lotfi Zadeh, George Klir, Merrie Bergmann, Arnold Kaufmann, Madan M. Gupta, Hung T. Nguyen, Elbert Walker, Petr Hajek, Jan Pavelka, Esfander Eslami, James Buckley, and Bart Kosko.

It isn't hard to find more if say one consults the bibliography of the Standford Encyclopedia of Philosophy on Fuzzy Logic.