Eh, I guess I have something I can contribute here.
Axiomatization
Axioms in mathematics are the basic foundations of entire bodies of mathematics. Though originally the concept of an axiom in mathematics was the same as in philosophy (a basic statement so simple and obvious that everyone will agree with it), its meaning has since morphed into something subtly different. Nowadays, axioms are the minimum requirements of any well-established body of mathematics (such as arithmetic, geometry, group theory, etc). Pretty much every theorem in that body of mathematics can be derived from those basic axioms (though sometimes we'll be in subsections of those bodies that require additional axioms).
There's some philosophical debate about what axioms actually ARE. In my opinion, they're the gatekeepers / minimum conditions of that body of math, the "if" of an if-then conditional assertion. If you are dealing with an application where the axioms hold, then all the theorems of that body of math can be used with deductive certainty. However, if you're dealing with an application where some of the axioms aren't true, then all bets are off. In a sense, the tools of math are only designed and guaranteed only for situations where their axioms hold, and using them elsewhere is a bad idea. This makes axioms a bit like a checklist that you run through before using that body of mathematics.
For example, one of the axioms of basic natural number arithmetic (kinda, I might qualify this in another post) is that if you keep incrementing a number (adding 1 to it), you will never loop around back to somewhere that you've been before, but will instead proceed through an infinity of new numbers. For example, if you start counting at 4 and go up through 5, 6, 7, and so on, no matter how long you go, you will never get back to 4. Now this might seem intuitively true in the sense of a philosophical axiom, but it's not that hard to find a situation where this axiom DOESN'T hold. Imagine adding (physically combining) one pile of sand to another pile of sand. You end up with a bigger pile of sand, but it's still ONE pile of sand. Here, 1 + 1 = 1. Does this mean arithmetic's false? Not at all. Instead, it violates the don't-loop axiom, and so it falls outside of the realm of arithmetic. It's akin to wanting to loosen a Phillip's-head screw, and reaching for a ball-peen hammer. You've chosen the wrong tool for the job... and it's your fault, not the tool's.
The first and arguably most-famous axiomatization was Euclid's treatment of geometry from sometime around 300 BCE, which he founded on five postulates. (Meaning axioms. Different name for the same thing.) This was back when an axiom was the same thing in math and philosophy, and Euclid was seriously putting them forward as axioms in the philosophical sense. He formulated all of geometry (as known in his day) as a series of logical deductions derived from these five postulates. In an example of intellectual integrity for the ages, Euclid openly admitted in Elements that he had a lot of doubt about the fifth postulate. He worked very hard to prove everything he could using just the first four postulates, and the stuff that depended on the fifth postulate he flagged as perhaps being unreliable. (The controversy around the fifth postulate is a mathematical epic spanning over two millenia that I won't try to summarize in this post.) Most of the things that Euclid published in Elements were already known, but Euclid's approached the material with a level of mathematical rigor that was nearly unheard-of. Plato's Academy would sport a sign reading "let none ignorant of geometry enter", because geometry after Euclid was seen as the benchmark of logical reasoning, while other areas of math such as arithmetic had not benefited from that rigorous treatment. (In time Greek philosophy would warp itself around geometry. Geometry came to be seen as noble-math and other fields as lesser. Concepts that could not be expressed in Euclidean geometry, such as nullity, negative numbers, the infinite, and the infinitesimal became derided and dismissed as impossibilities. Social philosophies came to be based on geometry, such as nobility being justified by comparison of large similar triangles to smaller ones.)
Let's jump ahead to Renaissance and modern times. Old doctrines were being questioned, including mathematical doctrines. 0 had been imported to Europe with Arabic symbols. Algebra was also imported from the Arab world by way of Spain. Descartes created Analytic Geometry, which essentially took most or all of geometry and boiled it down to algebra -- turning noble math into infidel peasant math. People started playing around with the infinite and the infinitesimal... and Leibniz and/or Newton invented calculus.
Calculus was in particular need of mathematical rigor because no one, not even its inventors, really understood HOW it worked. Newton thought of it in terms of "fluxions" (and the original name was "The Calculus of Fluxions"), where fluxions were variables that would become zero after some mathematical simplification was performed. The problem with this was that it was a cheat. Everyone knew that his technique had no basis in rigorous mathematics. One of his detractors would complain that he was making pacts with the ghosts of numbers departed. Yet at the same time, everyone could tell that his math WORKED.
Around the 19th century, with Calculus becoming increasingly important, other bodies of mathematics emerging, and geometries that called the Euclidean model into serious doubt, a big push was made to reestablish mathematical rigor. Most of our axiomatizations date back to either the 19th or early 20th centuries. The thing is, multiple mathematicians were working on this simultaneously, and they produced multiple different axiomatizations for most of the bodies of mathematics. I've seen about a half-dozen different ways to axiomatize real numbers, to name just one example. Which brings us to the last point I want to make about sets of axioms. In some situations these different axiomiazations lead to subtle differences and we have to be clear which we're working with. (For example, ZF versus ZFC in set theory. The difference between the two, the axiom of choice, is a bright candy shell of "oh obviously that has to be true" covering a sticky caramel center of "WTAF?!?!?!?") In other cases, different axiomatizations produce exactly the same results, even with different axioms. We call these axiomatizations equivalent, and we can prove that Axiomatization A = Axiomatization B by showing that every Axiom of A is a theorem of B and vice-versa. As a result, we can't really say what the axioms of any given body of math are... because most bodies of math have multiple, equivalent formulations of their axioms, and we can't really say that one is any more right than the other because they all amount to the same thing.
BTW, the 19th century's push for rigor ended spectacularly (by mathematician standards of spectacle) with Russel's Paradox. Or as clickbait would describe it, "How Bertrand Russel DESTROYED MATHEMATICS with ONE SIMPLE QUESTION!" But that's a story for another time.
Axiomatization
Axioms in mathematics are the basic foundations of entire bodies of mathematics. Though originally the concept of an axiom in mathematics was the same as in philosophy (a basic statement so simple and obvious that everyone will agree with it), its meaning has since morphed into something subtly different. Nowadays, axioms are the minimum requirements of any well-established body of mathematics (such as arithmetic, geometry, group theory, etc). Pretty much every theorem in that body of mathematics can be derived from those basic axioms (though sometimes we'll be in subsections of those bodies that require additional axioms).
There's some philosophical debate about what axioms actually ARE. In my opinion, they're the gatekeepers / minimum conditions of that body of math, the "if" of an if-then conditional assertion. If you are dealing with an application where the axioms hold, then all the theorems of that body of math can be used with deductive certainty. However, if you're dealing with an application where some of the axioms aren't true, then all bets are off. In a sense, the tools of math are only designed and guaranteed only for situations where their axioms hold, and using them elsewhere is a bad idea. This makes axioms a bit like a checklist that you run through before using that body of mathematics.
For example, one of the axioms of basic natural number arithmetic (kinda, I might qualify this in another post) is that if you keep incrementing a number (adding 1 to it), you will never loop around back to somewhere that you've been before, but will instead proceed through an infinity of new numbers. For example, if you start counting at 4 and go up through 5, 6, 7, and so on, no matter how long you go, you will never get back to 4. Now this might seem intuitively true in the sense of a philosophical axiom, but it's not that hard to find a situation where this axiom DOESN'T hold. Imagine adding (physically combining) one pile of sand to another pile of sand. You end up with a bigger pile of sand, but it's still ONE pile of sand. Here, 1 + 1 = 1. Does this mean arithmetic's false? Not at all. Instead, it violates the don't-loop axiom, and so it falls outside of the realm of arithmetic. It's akin to wanting to loosen a Phillip's-head screw, and reaching for a ball-peen hammer. You've chosen the wrong tool for the job... and it's your fault, not the tool's.
The first and arguably most-famous axiomatization was Euclid's treatment of geometry from sometime around 300 BCE, which he founded on five postulates. (Meaning axioms. Different name for the same thing.) This was back when an axiom was the same thing in math and philosophy, and Euclid was seriously putting them forward as axioms in the philosophical sense. He formulated all of geometry (as known in his day) as a series of logical deductions derived from these five postulates. In an example of intellectual integrity for the ages, Euclid openly admitted in Elements that he had a lot of doubt about the fifth postulate. He worked very hard to prove everything he could using just the first four postulates, and the stuff that depended on the fifth postulate he flagged as perhaps being unreliable. (The controversy around the fifth postulate is a mathematical epic spanning over two millenia that I won't try to summarize in this post.) Most of the things that Euclid published in Elements were already known, but Euclid's approached the material with a level of mathematical rigor that was nearly unheard-of. Plato's Academy would sport a sign reading "let none ignorant of geometry enter", because geometry after Euclid was seen as the benchmark of logical reasoning, while other areas of math such as arithmetic had not benefited from that rigorous treatment. (In time Greek philosophy would warp itself around geometry. Geometry came to be seen as noble-math and other fields as lesser. Concepts that could not be expressed in Euclidean geometry, such as nullity, negative numbers, the infinite, and the infinitesimal became derided and dismissed as impossibilities. Social philosophies came to be based on geometry, such as nobility being justified by comparison of large similar triangles to smaller ones.)
Let's jump ahead to Renaissance and modern times. Old doctrines were being questioned, including mathematical doctrines. 0 had been imported to Europe with Arabic symbols. Algebra was also imported from the Arab world by way of Spain. Descartes created Analytic Geometry, which essentially took most or all of geometry and boiled it down to algebra -- turning noble math into infidel peasant math. People started playing around with the infinite and the infinitesimal... and Leibniz and/or Newton invented calculus.
Calculus was in particular need of mathematical rigor because no one, not even its inventors, really understood HOW it worked. Newton thought of it in terms of "fluxions" (and the original name was "The Calculus of Fluxions"), where fluxions were variables that would become zero after some mathematical simplification was performed. The problem with this was that it was a cheat. Everyone knew that his technique had no basis in rigorous mathematics. One of his detractors would complain that he was making pacts with the ghosts of numbers departed. Yet at the same time, everyone could tell that his math WORKED.
Around the 19th century, with Calculus becoming increasingly important, other bodies of mathematics emerging, and geometries that called the Euclidean model into serious doubt, a big push was made to reestablish mathematical rigor. Most of our axiomatizations date back to either the 19th or early 20th centuries. The thing is, multiple mathematicians were working on this simultaneously, and they produced multiple different axiomatizations for most of the bodies of mathematics. I've seen about a half-dozen different ways to axiomatize real numbers, to name just one example. Which brings us to the last point I want to make about sets of axioms. In some situations these different axiomiazations lead to subtle differences and we have to be clear which we're working with. (For example, ZF versus ZFC in set theory. The difference between the two, the axiom of choice, is a bright candy shell of "oh obviously that has to be true" covering a sticky caramel center of "WTAF?!?!?!?") In other cases, different axiomatizations produce exactly the same results, even with different axioms. We call these axiomatizations equivalent, and we can prove that Axiomatization A = Axiomatization B by showing that every Axiom of A is a theorem of B and vice-versa. As a result, we can't really say what the axioms of any given body of math are... because most bodies of math have multiple, equivalent formulations of their axioms, and we can't really say that one is any more right than the other because they all amount to the same thing.
BTW, the 19th century's push for rigor ended spectacularly (by mathematician standards of spectacle) with Russel's Paradox. Or as clickbait would describe it, "How Bertrand Russel DESTROYED MATHEMATICS with ONE SIMPLE QUESTION!" But that's a story for another time.
Being an antipistevist is like being an antipastovist, only with epistemic responsibility instead of bruschetta.
Ignore list includes: 1 douche bag (Drich)
Ignore list includes: 1 douche bag (Drich)
