Why do mathematicians always agree?
Science is a lively social activity, with many claims being lively debated. What about mathematics? The cliché about mathematicians being poor at managing social relations is quite strong and widespread. One of the most famous joke on the topic goes like this:
Question: How can you spot an extrovert mathematician?
Answer: He looks at YOUR shoes when he talks to you.
Is mathematics “less social” than other academic disciplines? Some support for a ‘yes’ answer can be found in a recent piece of news. A famous mathematician, Nelson, had claimed to give a proof of a rather surprising proposition: “Peano Arithmetic is inconsistent.” Two other famous mathematicians, Tao and Tausk, said the proof included one specific mistake, which they spelled out. Nelson’s reaction was: “Ah, you’re right. So I have not proven that Peano Arithmetics is inconsistent”. End of the story. No fight, no disagreement, no formation of alternative schools of thoughts, no playing with how to interpret this or that claim. Just plain boring consensus.
(These are the axioms that Nelson claimed were inconsistent. They are supposed to express central propositions true of our system of natural numbers with addition. They are used to prove things about an object that is central in many cultures.)
Mathematics is full of that: easily achieved consensus. Everybody agrees. No debate, and yet, the consensus is not socially induced in any standard way.
In a recent blog post, The (in)consistency of PA and consensus in mathematics, Catarina Novaes takes this as a nice illustration of some points made by philosopher Jody Azzouni, who argues that mathematics is unique as a social practice.
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.
The third point needs further explanations. Azzouni’s argument is that, in mathematics, there has been no “drift” through small variations in interpretations. The social standard of what constitutes a proof has remained stable across time and so has the social practice of proving. This contrasts with the internalisation of, say, religious beliefs, where a different opinion can always lead to an alternative religion. Azzouni notes that « mistakes in mathematics are common, and yet mathematical culture doesn’t splinter because of them, or for any other reason (for that matter); that is, permanent competing practices don’t arise as they can with other socially-constrained practices. This makes mathematics (sociologically speaking) very odd. »
I think that the research question is well put: doing mathematics is a social practice, but it differs in important ways from most other social practices. Mathematics is learned, communication plays a crucial role in its maintenance and development, mathematicians form a distinct community with its institutions … and there is a strong, puzzling, homogeneity of opinion. (I am not convinced that standards of proofs did not evolve and that debates in mathematics had little impact on the practice itself, as Azzouni suggests; but it remains that the ease through which uniformity of opinion is achieved is puzzling).
What causes this homogeneity? It is mainly when one thinks of sociology as limited to analyses of power relations — of how groups of people impose beliefs and practices on others, that social investigation is deemed irrelevant (giving way to Platonism or psychologism). If, by contrast, social investigation means looking at all the factors that might produce consensus and homogeneity of opinion, then a social and naturalistic investigation can be carried on. Thus the slogan: when facing the specificity of mathematics, don’t put sociology out; but put psychology in. In particular, we can hope that psychologists will have something to say about why mistakes are so easily acknowledged. What happens, for instance, with the confirmation bias? Why are evaluations of proofs generating no controversies, while evaluations of scientific arguments do?
12 Comments
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Christophe Heintz 3 December 2012 (10:32)
Answer to apG palma:
You are right that there is substantive disagreement about what mathematics is about. There has also been debates about specific mathematical notions (e.g. irrational numbers, infinitesimals), and, to a lesser extent, about how proofs are to be done or presented (e.g. use of diagrams, the excluded middle or computers in proofs).
I acknowledge the existence of these debates. My observation was restricted to consensus over proofs: there is little debate about whether a proof is correct or not. Of course, once a proof is published, mathematicians do check it. Yet, they rapidly come to an agreement about its correctness. There are exceptions: some mathematicians might be reluctant to accepting a proof because of the notions or the method it relies on. And it may take some time before someone shows that a proof is mistaken (e.g. 11 years for showing that Kempe’s 1879 pseudo-proof of the four-colour theorem was mistaken). Still, a noteworthy degree of agreement among mathematicians that I and some philosophers of mathematics such as Azzouni find puzzling and thus worth further investigation.
You also suggest that the ontology of mathematics could provide some elements of answer to why mathematicians agree so frequently. And indeed, I guess that ontological theories were develop with this striking agreement in mind. But if Platonism (mathematics is about unworldly entities), formalism (mathematics = axioms for sets + a few intuitive inferential steps) or psychologism (the idea that mathematical truth are already in our mind and only need to be extracted) have a story about the origin of mathematicians’ convictions, I still think none of them provide a satisfactory explanation as to how mathematicians end up making mistakes and yet agreeing so often.
Martin Stehberger 5 December 2012 (16:31)
This is a very interesting question, and thanks also for the link to the recent high-calibre example case. The agreement found in the mathematical community is unusual. Still, I would take it as conceptually the standard case, and try to determine why other academic disciplines “fail”. Two thoughts (admittedly simplified, and I don’t want to distract with specific examples):
Science: Experimental data allow ambiguities and wiggle room so that confirmation bias, unwillingness to admit mistakes, etc can play their roles. However, if the discipline is a proper science then with more specific, complete and reliable data there is typically consensus later on.
Philosophy: Here, my best guess would be that philosophy (where it tries to establish truths, like science and mathematics) consists today mostly of those areas that are affected by clashes between strong intuitions and rigorous rational thinking. Consensus in favour of the intuitions would not be stable against the rational challenge; but the rigorous thinkers on the other hand haven’t managed, against the headwind of intuition, to achieve critical mass within their fields to monopolise them. Standards haven’t been established as they have been in mathematics. Surely there are “proofs” of famous unsolved mathematical theorems by “crackpots” who insist on the validity of their proofs, but that doesn’t count as violation of the mathematical consensus.
Dan Sperber 6 December 2012 (16:16)
I like the way Martin reframes Christophe’s question. 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.)
I am not too enthusiastic about such a Cartesian answer to the problem. Are you? It you too are not, it would be worth trying to spell out what is missing from it.
Christophe Heintz 10 December 2012 (23:22)
@Martin
Your way to re-frame the question is akin to the way it was framed by Merton or Lakatos: scientific development is a rational enterprise, but sometimes, some factors others than rationality come in. What stands in needs of explanation is only the non-rational part of it. Inasmuch as this approach dispense historians of science from giving causal explanations of most scientific events (viz. those that we now deem rational), it is unsatisfactory.
Concerning scientific debates: I think it is a bad idea to categorise a priori those who won the debate as rational and those who lost it as irrational. Opponents to true theories and proponents of false theories are very often *not* stupid, irrational, biased or victims of strong intuitions that go against rigourous rational thinking. They have had their own reasons. Phlogiston theorists, Ether theorists, defenders of the Ptolemaic system after Gallileo, etc. had all good reasons to believe what they believed.
In science, you have many cases where two intelligent, serious and informed expert scientists hold beliefs that are inconsistent one with another. It seems that it is less the case in mathematics. Saying that irrationality has less hold on mathematics than in other disciplines is, I believe, unsatisfactory because most scientific debates are not resulting from one party being irrational.
Christophe Heintz 10 December 2012 (23:25)
@Dan
Relevance as you defined it is relevance of an input to someone; it varies in view of background knowledge and interests of the person. The relevance of scientific practices and theories can vary across scientists, since they have had, as human beings, varied personal histories and interests. Kuhn pointed out that for a change of paradigm, there often need to be a change of generation because the old generation having been trained in the old paradigm too easily interpret data with the old theoretical framework. Kuhn also described how a change of paradigm is slowly brought up within normal science, through giving more and more importance to anomalous results: slow changes in the distribution of relevance. To me, interest theory of the Strong Programme is to be interpreted exactly as pointing out that theories might have more or less relevance to different social groups.
So it does seem to me that scientific disagreement can in part be explained in terms of the variable relevance of a theory to the diverse scientists. The notion of paradigm, then, explains why disagreement is strongly tuned down during “normal science”: paradigms are common cognitive environments that equalize relevance across actors. But how happen disagreement is so much tuned down in the case of mathematics? And what means are used to make attention converge that would be proper to mathematics?
Martin Stehberger 13 December 2012 (14:51)
First of all, thank you for clarifying that proponents of false scientific theories generally don’t have to be irrational or biased. Indeed, I have no reason to doubt the merits of phlogiston or ether theory relative to what was known in their respective times.
What about scientific debates? Empirical data can be ambiguous; due to this, I said, “confirmation bias, unwillingness to admit mistakes, etc can play their roles”. But you’re right, that’s not the whole story. When the data are genuinely consistent with different versions of reality, and different scientists fill the roles of their advocates, I wouldn’t want to call anyone “irrational”.
Whether irrationality is involved or not, though, we can still attribute the lack of consensus to ambiguity in the data. By contrast, mathematics doesn’t use empirical data. That’s now where the comparison to parts of philosophy, also “pure thinking in search of truths”, but without consensus, suggests itself. How to understand the difference between mathematics and these parts of philosophy? My crude answer was that mathematics has managed to establish sufficient standards of rigour whereas philosophy still hasn’t, the reason in turn being that philosophy is affected by clashes between intuitions and clear thinking that have (so far) no equivalent in mathematics. Trying to translate reasoning into Dan’s comparison case of vision:
Science: blurred vision (but each image becoming over time less blurred);
Philosophy (parts of it; today): vision interfered with by preconceived images.
In reality, these two anti-consensus sources are not cleanly separated by field: science can also be counterintuitive as philosophy should be; philosophy can also use empirical data as science does. Also, philosophy (parts of it) is not just a duel between certain intuitions and clear thinking, since the lack of standards can leave openings for all sorts of fuzzy thinking.
I liked the joke about the extrovert mathematician. The relationship between mathematics, science and philosophy is also the subject of a famous joke (the one with the waste-paper baskets): http://consc.net/misc/mpp-joke.html
Christophe Heintz 19 December 2012 (14:06)
@Martin
I agree with you when you point out some sources of disagreement in empirical sciences and philosophy. What is less clear to me is why these sources have less effects on mathematical practice. How do mathematicians avoid conflicts of interpretations? How do mathematicians recruit or dismiss intuitions when proving? Nothing is obvious here: mathematicians as well as empirical scientists do have to interpret: they have to interpret notions that are far from obvious such as ‘infinitesimals’ and ‘polyhedron’. And Mathematicians as well as philosophers have to tackle with intuitive and unintuitive aspects of their thinking: irrational numbers posed problems to the Greek because counter-intuitive; and try to think of a torus in a five dimensional space!
What I therefore suggested is that in order to answer the question why mathematicians agree when it concerns proofs, one needs to consider the cognitive practices of proving. The cognitive practices are both a cultural phenomenon (they are shared practices and somewhat specific to the community of mathematicians) and a cognitive one (the practices are meant to deal with representations and they concern mental processes). One of their effect is agreement in spite of the fact that mathematicians’ discourse are complex.
Christophe Heintz 19 December 2012 (14:12)
@Catarina
Thanks a lot for the clarifications. My goal when posting to that blog was not to add something to what you say in your blog, but only to attract the attention of people working in cognition and culture to the strange cultural phenomenon you’ve put in light. The interesting idea, which is present in your post and Azzouni’s paper, is that agreement in mathematics is a puzzling cultural phenomenon and that an apparent explanans will be of a psychological nature.
I continue recommending the readers of this blog to have a look at the comments on yours, especially if they are interested in pro-formalism type of answers to the question (viz. mathematicians agree because they apply a simple set of deductive rules).
Dialogical reconceptualisation of deduction: no doubt it will please Dan Sperber and Hugo Mercier, who developed the argumentative theory of reasoning. Netz, in The Shaping of Deduction, suggests that mathematics indeed originates in an argumentative historical setting prompting the need for alternative argumentative practices. Thinking of proving as a specific type of argumentative practice is historically enlightening. It is also comforting, if one buys the argumentative theory of reasoning, and it opens a research question: to what extend the reasoning capacities as they are rediscribed in the argumentative theory of reasoning constrain and enable the cognitive practice of proving? I’m looking forward to reading more on your hypotheses. At the moment, I still fail to see what built-in opponent features lie in mathematical practice but not in other practices less successful in achieving agreement.
Martin Stehberger 30 December 2012 (23:20)
Interpretation problems also in mathematics?
Your example of infinitesimals made me see this point. I don’t know the exact history but could imagine that at some point mathematicians realised that some such notion was needed, while disagreeing on its exact shape. So, there would have been interpretation problems a bit like in empirical science.
A difference is that such disagreements in mathematics, at least today, will be not about what is *true* but about the best way to advance the field. If we just look at proving, I don’t see substantive interpretation problems any more. In a modern mathematical proof, a notion like polyhedron may not be obvious — in the sense of “easy to grasp” — but it will be rigorously defined at the outset. And if a mathematician uses a new notion in a proof, he/she will have to define it first (running the risk that the definition will not catch on with other mathematicians because the notion could have been interpreted better or turns out to be unimportant anyway). Ambiguous objects are not allowed, whereas empirical science cannot avoid dealing with ambiguous empirical data.
(You might ask: exactly how does mathematics keep ambiguous objects out? How does it ensure unambiguous “blueprints” for difficult-to-grasp objects? Certainly the nature of the standards of rigour in modern mathematics is not a trivial issue. But my focus here, starting with my initial reframing of your title question, was not on how mathematics succeeds but on what makes it hard/harder for science and philosophy to do so.)
Treacherous intuitions also in mathematics?
I agree that irrational numbers or a torus in a five dimensional space are counterintuitive. A corresponding treacherous intuition would be such a torus cannot exist. This type of intuition could indeed lead to disagreement, but I would think that in today’s mathematics, with its abundance of abstract formal constructions of new objects from existing ones, it has perhaps lost its sting.
For the sort of treacherous intuition I had in mind for philosophy (when I said “clashes between intuitions and clear thinking”), the nearest mathematical analogon I can come up with is the set of all sets. In naive set theory one can form this object and then investigate further; e.g. one could consider the subset of all those sets that don’t contain themselves as a member, and ask whether this subset contains itself as a member … contradiction! Russell’s paradox.
If some mathematicians still insisted on their ability to form the set of all sets as an abstract object, they could wreak havoc using it together with ordinary methods of proving. Fortunately, in response to Russell’s paradox, mathematics has introduced safety standards for set-forming, ensuring that no set can be a member of itself; and that seems to work well. But now imagine firstly that the contradiction was harder to present (a longer chain of deduction and/or the meaning of “set” or “member” being less clear than it really is), and secondly that this wasn’t an isolated incident but that the field was overflowing with such cases. In parts of philosophy, I think, such a situation is ultimately responsible for today’s lack of consensus.
For example, look at the first page of Dennett’s commentary on Fodor’s paper “Against Darwinism”. They are both eminent analytic philosophers of mind, but far apart. Dennett says that Fodor’s argument really does follow from the premises as far as he can see, so it’s not the practice of proving that separates the two here. Instead he claims to have found no less than four false premises. (Although I should add that I don’t claim this to be a principled distinction in general: sometimes you can redescribe a fault in a proof as “the author invokes a false premise here”.)
Is proving specific to mathematics?
Some deduction also occurs in daily life, as in, say, “we know it’s in the room, but it’s not in that drawer, so it must be …”. The more formal concept of “proof” is still not unique to mathematics, think of judicial trials. Perhaps the cognitive practices of proving appear “somewhat specific to the community of mathematicians” (as you put it) simply because this community has made such a success of them?
Then again, one might turn that round and try to *redefine* modern mathematics somehow as “that which results if you systematically build abstractions on them, avoiding notions that turn out to be unsafe for this purpose, however useful in other respects”. (E.g.”number” seems safe, “propositional attitude” unsafe.)
William (Bill) Lawless 29 March 2013 (19:14)
Dear learned scholar of mathematicians, I disagree with your premise that mathematicians do not disagree, and, being wonderful souls, are easily converted to consensus. No less a scholar, intellectual and role model than Von Neumann (1961), the founder of game theory, argued against your premise. In fact, he bemoaned that unlike physicists, mathematicians who don’t agree behave in an unsocial manner by striking out in new directions, leaving their conflicts unresolved. In his article, the first in his collected works, Von Neumann wished that mathematicians disagreed as physicists did. Whenever conflict arose between two physicists (e.g., Bohr and Einstein), physicists refused to ignore it, often bringing their field to a standstill until a resolution was found (i.e., consensus via debate, unlike your fanciful example of consensus without debate). I have long cherished Von Neumann’s insight, and his remarkable paper on mathematicians. BTW, in my research, I too have found that consensus without conflict is indeed possible, except that none of the participants can agree on the result.
Von Neumann, J. (1961). The mathematician. Collected works, Pergamon.
Dennis Augustine 12 January 2014 (22:30)
I think they tend to agree more than others because they begin with strong agreement on very precise fundamental definitions and most of the rest follows logically and almost mechanically; which is not to say that there are not leaps of genius and intuition but that even the most once such a leap occurs those who follow can easily bridge the chasm using well defined steps.
The less consensus there is on fundamental definitions the less consensus there will be in any discipline. Thus we find mathematics (where consensus on fundamental definitions is strong) exhibiting strong agreement elsewhere, less so with physics, less with biology, less with history, less still with sociology, psychology, philosophy and so it goes. In a field like mathematics there the fundamental constructs are very simple and well defined (e.g. integers, vectors, operations, etc), but on the other end of the spectrum in a field like philosophy, the most fundamental constructs are perhaps the least well defined and agreed upon (e.g. reality, knowledge, truth, etc.) The level of consensus in each field is directly related to the precision of their fundamental definitions and their agreement with regards to them.
Weshah Razzak 18 August 2014 (05:42)
Great stuff. Is it because of rules? Everyone in the field knows the rules. Math is logic. Its also a language. Errors are spotted and corrections are agreed upon.