[As an aside, I first want to say that I really want to start using gender-neutral pronouns. I find he/she, (s)he, and alternating between he and she to all be awkward or ridiculous. I think it's most reasonable to use just one pronoun when referring to "a generic person", and "they" simply won't do since it's plural. So, I shall begin using gender-neutral pronouns. There are many conventions out there, and as far as I am aware, none is more widely adopted than the other, so I will use the one I think sounds best, which is the Spivak convention (ey/em/eir/eirs/eirself).]
When a mathematician has a theory (usually called a conjecture), ey sets about to prove it using all of the tools of logic available to em. If ey believes that ey has constructed a convincing proof, ey proceeds to share it with peers. It is then reviewed by peers who determine whether they believe the proof or not. If a hole in the logic is found by anyone, it is presented and the original mathematician (or possibly others) try to fix the hole, if it can be done. Sometimes this is a very simple process, where no revision is necessary, and other times it can be a very complicated and drawn-out process, taking several months or several years to complete.
Even after the paper is published, it is not immune to scrutiny. It has happened many times that after publication, something has been found to be wrong with a paper, and another mathematician (or the same one as the original author) will write another paper indicating the error of the previous. This is good for the mathematical community because it helps minimize the amount of error that is propagated.
Why is it so important to have correct mathematics? Because every science is based on mathematics. With incorrect or flawed math, all of the applied fields--biology, chemistry, physics, engineering, computer science, etc--would also be flawed. Correct mathematical principles are necessary for understanding the world in which we live.
Imagine, for a moment, if mathematical truth were treated the same way as religious truth. One mathematician would make up rules for math that ey thought was true. Ey would then claim that a superior being revealed these mathematical truths to em through some supernatural form of communication. Ey would then preach these ideas of math to other people. Ey would ask other people to learn these principles and to accept them. Perhaps ey would encourage others to pray about these principles to find out if they are true--not to scrutinize them and determine whether there is a flaw, but just to get a warm feeling in eir heart that these teachings are in fact true. The actual veracity of the teachings would not be on trial here, but only the learner's will--to see whether it would bend to that of the teacher.
Imagine that this original teacher actually did have a flaw in eir mathematical principles somewhere. As society grew and more people adopted this flawed system, the other sciences would suffer from it. Rocket ships wouldn't be able to make it to the moon with any real accuracy, because the math being used to try to get them there would not be working. People might start asking whether the math behind the rocket theory was correct, to which the math-prophet would reply that they simply needed to have more faith in the mathematical teachings, and try harder to accept them into their own lives.
The error would never be solved, because it would not be suspect of causing this harm (or impediment) in society. Anyone who might suggest that ey had a better version of the truth would be told that only the math-prophet was capable of receiving mathematical revelations, that the math-god knew all mathematical truth and would never deceive his people. People who did question the math would be labeled as heretics, infidels, apostates, and heathens. Their opinion in anything would instantly not matter. Simply for questioning a mathematical principle, they would be cast out of society and hated.
I hope that I have presented sufficiently the case for critical thinking and logic in the interest of seeking truth. The only way to know the truth is to question what we currently accept as truth, to determine whether it really is true. We must attack it at every angle. It must never be preserved by words such as "sacred", "holy", or "divine". These words only allow errors to be multiplied and perpetuated. Question why something is true, question how it is true, question what evidence would support its truth, or what would its truth or falsehood imply. Don't just accept something and drown out all contradictions. That way you learn nothing. That way you are arrogant. Admitting that you might be wrong is the only way to attain a higher level of truth.
When a mathematician has a theory (usually called a conjecture), ey sets about to prove it using all of the tools of logic available to em. If ey believes that ey has constructed a convincing proof, ey proceeds to share it with peers. It is then reviewed by peers who determine whether they believe the proof or not. If a hole in the logic is found by anyone, it is presented and the original mathematician (or possibly others) try to fix the hole, if it can be done. Sometimes this is a very simple process, where no revision is necessary, and other times it can be a very complicated and drawn-out process, taking several months or several years to complete.
Even after the paper is published, it is not immune to scrutiny. It has happened many times that after publication, something has been found to be wrong with a paper, and another mathematician (or the same one as the original author) will write another paper indicating the error of the previous. This is good for the mathematical community because it helps minimize the amount of error that is propagated.
Why is it so important to have correct mathematics? Because every science is based on mathematics. With incorrect or flawed math, all of the applied fields--biology, chemistry, physics, engineering, computer science, etc--would also be flawed. Correct mathematical principles are necessary for understanding the world in which we live.
Imagine, for a moment, if mathematical truth were treated the same way as religious truth. One mathematician would make up rules for math that ey thought was true. Ey would then claim that a superior being revealed these mathematical truths to em through some supernatural form of communication. Ey would then preach these ideas of math to other people. Ey would ask other people to learn these principles and to accept them. Perhaps ey would encourage others to pray about these principles to find out if they are true--not to scrutinize them and determine whether there is a flaw, but just to get a warm feeling in eir heart that these teachings are in fact true. The actual veracity of the teachings would not be on trial here, but only the learner's will--to see whether it would bend to that of the teacher.
Imagine that this original teacher actually did have a flaw in eir mathematical principles somewhere. As society grew and more people adopted this flawed system, the other sciences would suffer from it. Rocket ships wouldn't be able to make it to the moon with any real accuracy, because the math being used to try to get them there would not be working. People might start asking whether the math behind the rocket theory was correct, to which the math-prophet would reply that they simply needed to have more faith in the mathematical teachings, and try harder to accept them into their own lives.
The error would never be solved, because it would not be suspect of causing this harm (or impediment) in society. Anyone who might suggest that ey had a better version of the truth would be told that only the math-prophet was capable of receiving mathematical revelations, that the math-god knew all mathematical truth and would never deceive his people. People who did question the math would be labeled as heretics, infidels, apostates, and heathens. Their opinion in anything would instantly not matter. Simply for questioning a mathematical principle, they would be cast out of society and hated.
I hope that I have presented sufficiently the case for critical thinking and logic in the interest of seeking truth. The only way to know the truth is to question what we currently accept as truth, to determine whether it really is true. We must attack it at every angle. It must never be preserved by words such as "sacred", "holy", or "divine". These words only allow errors to be multiplied and perpetuated. Question why something is true, question how it is true, question what evidence would support its truth, or what would its truth or falsehood imply. Don't just accept something and drown out all contradictions. That way you learn nothing. That way you are arrogant. Admitting that you might be wrong is the only way to attain a higher level of truth.
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Anyone is allowed to comment on this blog. As you can tell from reading my blog, I am very opinionated and I'm not afraid to share my opinion. You're welcome to disagree with me as mildly or vehemently as you like, but be aware that I will reply with my own opinions, very strongly. If you don't want that kind of open discussion, or you think it will hurt your feelings, then please avoid posting. I do try to be respectful, but my verbology often comes across as brusque.