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Given access to the same facts, how is it possible that there can be disagreement between experts in a discipline in Chemistry and History. Which kind of examples have been seen.

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With respect to math, keep in mind that - although we do have a notion of "fully formal" (and in particular, computer-verifiable) proof, mathematicians rarely actually write such proofs. If you look at a mathematics paper you will almost certainly find arguments in (admittedly very jargon- and symbol-filled) natural language. There is therefore lots of room for error here: although we think of a "rigorous informal" argument as a blueprint for a fully-formal proof (and actually more than that: an intuitive explanation as well), there may be gaps in the argument which are not obvious. For example, consider the current situation with regard to Mochizuki's claimed proof of the ABC conjecture.

This is ignoring more subjective points, which do exist in mathematics. For example, mathematicians might disagree about the plausibility of a conjecture, or the importance of a topic, or the validity of a logical framework as a foundation of mathematics.

Ultimately, mathematics - at least as practiced by people (as opposed to what one might argue it should be) - is more than simply the performance of calculations, or even the discovery of formal proofs. As a human endeavor it is also, and to be honest far more, about understanding.


Meanwhile, I don't see where you're coming from re: art at all. What's an example of a question about art which isn't highly subjective? (The only things I can think of are questions about art history, e.g. "Who painted the Mona Lisa?," but that seems separate from art questions.)

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  • Out of curiosity, why the downvote? Mar 18 '21 at 5:15
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Judgments about art are about as subjective as you can get, so it's not surprising you can have disagreements.

When it comes to Mathematics, it's easy to disagree on how pieces of Mathematics fit together. For example, it's not uncommon for certain conjectures to be believed by some and not others. One reason is that people try to understand different things based on intuitions and paradigms that vary significantly depending on one's training, subfield, etc.

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Within foundations of mathematics, there are different views how mathematics should be treated ontologically

  • Formalist, which sees mathematics as a game of manipulating symbols.
  • Platonist, which views mathematical objects as transcendent objects with an abstract existence of their own and formal systems merely as tools to investigate them.
  • Realist, an extreme form of platonism which views mathematical objects as fundamental building blocks for our reality.
  • Intuitionist, they reject the law of excluded middle and view mathematics as a subjective discipline without ontological importance.
  • Constructivist, intuitionist in method but agnostic about the ontology of mathematics

and a few others. These approaches can influence which axioms and (inference rules) are set. For example Intuitionist will reject the law of excluded middle as axiom and additionally, in order to prevent it from being derived, they also have to remove the law of double negation from the axiomatizations of propositional logic. The axiomatic system determines which theorems can be derived. Thus the foundational views on mathematics can be one cause of disagreement between mathematicians. This is also often seen in interpretations of Quantum Mechanics where the ontological state a physicist will assign to the wave function depends on the interpretation of QM and thus in extension on philosophical questions about mathematics. The interpretation of a mathematical theorem, especially in context of physics, is often subjective and thus varies widely based on the view of the respective mathematician. Most mathematician are Platonist with a formalist working approach and do not work with formalalized proofs very often. For complex proofs of a theorem, there is often more time needed in order to figure out whether the proof is correct or not.

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