The Subjectivity in Mathematical Proofs

About 400 years ago, the French mathematician Pierre de Fermat left the world a famous riddle, which we now call Fermat's Last Theorem. It took us until 1994 to prove it, by Andrew Wiles, and another year to reprove it, by the same person, after a small mistake was detected by a peer. Which is curious: if mathematics is considered objective, how can two people have two different interpretations of the same result? If for Wiles the theorem was proven, in an objective field, we should suspect that the same conclusion could be reached by everyone else, regardless of the background. What made him think the proof was finished? Who decides where to stop?

We often hear praise for the objectivity of mathematics. The most idealist even suggest that everyone should use more mathematics and less intuition because one is closer to the Truth and the other leads to bad decisions, often known as cognitive "biases". To be able to question this notion of objectivity in mathematics we need to clarify what kind of math we are talking about and from what perspective we are viewing it. There are two main ways to look at the field. From one perspective, we can see the mathematics that is objective by construction in the way we apply particular formal rules to particular formal objects, as if we were playing chess but in a more symbolic and abstract world. But that is not the only form of mathematics that we can encounter in the wild. Outside this sterile and formal world, lives the kind of mathematics that Andrew Wiles and other notable mathematicians see. In this different dimension they create new definitions for new concepts, they use the rules of the field in a way that the strictness of logic cannot grasp - they create. This is where we can find the usual mathematical proofs that trouble the greatest minds. Away from the Gods of Plato and Aristotles, we find mathematical proofs in the much more subjective and nuanced universe of human activities. This distinction is obvious to the attentive practitioner of mathematics; however, we tend to miss it, even at the graduate level, the longer we dedicate ourselves to superficial mathematical work¹.

Mathematics was ontologically detached from logic for most of its existence. Math was not a derivative of logic, and logic was also not a derivative of math. Even with the parallel existence of Aristotelian formal logic², proofs were written and accepted without requiring any special logical-deductive formalism. With the works of Frege and Russell in the late 19th century, mathematics started to turn into a more Logical field³. The hope at the time was that mathematics could be reduced to a set of symbols and theorems, all derived formally from axioms. These theorems could also be algorithmically verified, ending the uncertainty and subjectivity in mathematics. Only if we could reduce everything to symbols and finite, unambiguous, rules. Either we solve the problems as abstract objects in an abstract world full of pure formalism and symbols and leave them there⁴ , or, if we want to see some actual meaning from those theorems and results, we need to translate them back to the real world by paying the price of loss in objectivity. However, the attempt of converting mathematics into pure logic has been grossly abandoned ever since.

In the real world, mathematics, as it is practiced, is subjective and dependent on the shared intuition of the current community that composes the field. But to say that it is subjective is not to say that it is arbitrary. The theorems and propositions still need to be fairly consistent in their applicability; they need to be understood by the community.

In 2012, Shinichi Mochizuki published an indecipherable proof of the ABC conjecture. His proof contained about 500 pages of new formalism that not even experts in the field could completely follow. Despite the author's perception of a finished proof, experts cannot confirm or falsify the proof. Mochizuki is part of the community that should be able to verify his proof. Therefore, if this is a truly objective field, it should be independent of at least of the expert that validates it.

Assuming that Mochizuki is an honourable person who doesn't want to trick everyone else—he is also reputable in his field and has been working on the problem for 20 years—we have to make him a relevant element of the community he is part of. Therefore, if math is objective and he is able to validate himself as an expert in his own field, the result should be verifiable by everyone else in the community. Since it requires background and an appropriate level of skill in reading, it is, in fact, a subjective endeavour. Or does truth in Mathematics depend on the number of experts that approve it? Then what is the number of people that need to accept the proof for it to be officially accepted? 

The moment we need to describe propositions and theorems in common language is the moment we lose symbolic and formal objectivity. The problem of using common language and loosely defined symbols to describe factual entities is that there is always an interpretation narrow or literal enough that it turns out to be true and also a sufficiently wide interpretation such that it turns out to be false, or vice versa. For example: In Euclidean geometry, the parallel postulate asserts that given a line and a point not on the line, there is exactly one line parallel to the given line that passes through the point. However, in non-Euclidean geometries like spherical or hyperbolic geometry, the concept of "parallel" differs. In spherical geometry, no parallel lines exist because all lines eventually intersect. In hyperbolic geometry, there are infinitely many lines through a point that do not intersect the given line and are considered "parallel." Therefore, the truth of the parallel postulate varies based on the interpretation of "parallel" within different geometries. In this example, we have a clear picture of the pluralism in the interpretation of mathematical definitions and axioms.

But for every one of these cases, where the difference in scope is more or less clear and accepted, we have an infinite number of them where the effects of different contexts and interpretations in mathematics are not so clear. There are theorems where two experts might think they have the same interpretation until a very specific counter-example is presented and in one of the expert's mind it makes sense to accept the counter-example as consistent with the theorem and from the other expert's perspective, one should reject the theorem because the counter-example refutes it. In the end, mathematics, as with any other human endeavour, doesn't offer us absolute truths; it gives us good enough truths that can help us live our lives a little better.

The proof of a theorem is just a means to convince others to accept the theorem. Failing to see the role of persuasion in how we do mathematics might be the shortcoming of Mochizuki in the example above. For the proof to be considered valid, he needs the acceptance of the community; he depends on this subjectivity. But maybe he can't make the proof more understandable due to the nature of the problem itself or simply because of the way the proof went off from the start. Contrary to common belief, when doing mathematics, we first have an intuition about a solution; only then are we able to prove it. The intuition precedes the proof. The sequence of steps that compose a proof acts only as a post-rationalisation that guides us in improving the soundness of our intuition. 

One of the biggest challenges for a practicing mathematician is how to live with and work with this pluralism of perspectives. One can be pragmatic by understanding how mathematics is actually practiced and, at the same time, developing the expertise to manipulate and work in this symbolic and abstract world. Facts, as we use them, are actually objective, but they are also just true propositions, abstract objects that, by themselves, are unable to give us direct consequences for the real world. We need to interpret these facts and use them in further arguments, away from formalism; there we have subjectivity again. When a government publishes their country's annual change in GDP, there is in fact objectivity in how you compute the rates of changes as percentages; there is an exact formula for that. However, there is a world of interpretations that can't be derived logically from these formulas. For example, what counts as part of GDP or not is a matter of international agreement; what an increase or decrease in GDP means for the economy and the livelihoods of people depends a lot on what sectors drove that GDP increase. In this case, we have an objective calculation and mathematical procedure, but at the same time, it is meaningless by itself. It is the interpretation of facts that has the means to establish causality and consequence.

The more certain and objective something is, the more meaningless it gets; meaning and objectivity usually go in opposite directions. That is perhaps why reading a book, contemplating a painting, or listening to a symphony, due to their inherent subjectivity, can be the most meaningful experiences we can have as humans.

1. Such as Economics, Psychology, Climate, Medicine and some Computer Science.

2. True and False, modus ponens and the Law of the Excluded Middle.

3. Their works also inspired Logical Positivism movement within the Vienna Circle during the 20th century, especially Wittgenstein, a student of Russell's.

4. See proof theory and model theory.